If that's true, then the bill/law will be thrown out. I don't think it's true, though, if for no other reason that some of it is obviously true. The President can't sign a bill unless it is passed by both the House and Senate, identically. That's obvious, and I can't imagine the Supreme Court ruling otherwise.
I suppose the question might be what happens if the House votes on a bill, and the Senate votes on a bill, and the President signs that bill into law, but someone determines that the process by which the Senate voted violated Senate rules? Is the law overturned? I don't know if it has ever been tested, but my guess is that it is not. I think the court would rule that the Senators knew what they were doing, and met the Consitutional requirements.
I also think that various commentators might have lost track of exactly what's going on. There's the budget reconcilliation process, that has been referred to as "reconcilliation", and that has been talked about as a way to bypass the filibuster, but there is also a different process by which two similar, but non-identical bills get "reconciled" between the two houses. I think that can also be used to bypass debate, and I think that's what is being looked at now. I'm not certain about this. I haven't tried to become an expert in Senate procedure.
It is budget reconciliation that is being proposed. That is why the Byrd Rule is being mentioned.
See my update, though. There are conflicting reports on whether the Senate Parliamentarian really said that the bill must be signed before reconciliation. I trust that he knows the difference between conference committees and budget reconciliation. And conference committees
never happen after a bill is signed. If he said it, then my analysis you quoted is completely correct. But my analysis depends in part on his having said that. If he didn't say it, then it may be the case simply that the Senate bill must be signed before the reconciliation bill is
signed. But even that requires the Senate bill and the Reconciliation bill to be two separate bills.
Even if the Parliamentarian didn't say that the Senate bill has to be signed before the Senate can do budget Reconciliation, I'm still pretty sure the exact same bill has to have at least
passed both houses before the Senate can do Reconciliation, because as you pointed out, budget reconciliation is not the same as conference committee, where different House and Senate bills are combined into a compromise bill. So the question is whether whatever passes under the Slaughter rule is really the same bill as the Senate bill. Even if the Huffpo is right about what the Parliamentarian said, the Senate bill and the reconciliation bill still have to be signed
separately, even if they can both be passed before either is signed.
So the question will be precisely whether what the House has passed through the Slaughter rule is two separate pieces of legislation. My position is that it would be a single amended bill. The House Democrats would contend that it is two separate bills: the original senate bill, exactly as it passed the Senate, and a separate Reconciliation bill. If it really counts as two bills, then there is no bicameralism problem. But the final product of deeming has always been one bill, with a "deemed" amendment included. The "deeming" merely means that the vote for the bill and the vote for the amendment are the same vote. The product is still a single piece of legislation. It isn't clear whether it is a simple matter of House rules, which the House is allowed to make. The House can pass two things with one vote if they make a rule to do so, but they are pushing the limits when they say that the result of that vote is two separate bills rather than one.
That narrow question would decide the matter. If it is two bills, then they have passed the Senate bill. The Senate could then proceed with budget reconciliation, either before or after the Senate bill is signed, depending on how the Senate Parliamentarian rules. If it is a single bill, then the House has passed something other than the Senate bill. This would preclude budget reconciliation in the Senate, and Senate Republicans could filibuster if the Senate takes up the combined bill+fixes as a single bill. If the product of the Slaughter process is two separate bills, then the House would have passed the same bill the Senate passed, and the Senate could take up reconciliation.
There are two bills, the Senate bill and a reconciliation bill. Both have to be passed by both Houses, and they have to be separate bills, because the former must either be signed before the latter is passed, or signed before the latter is signed, depending on what the Parliamentarian says.
What it comes down to, for me, is that if everyone does indeed vote on the same bill, and it passes with a majority in both houses, then I'm not at all concerned about how it got there. If, on the other hand, they try to avoid voting on the exact same bill, but have the President sign it, I'm equally confident that the Supreme Court will call foul, and so Congress won't even try it. I guess we'll see.
The question will be whether or not the Senate is entitled to proceed without having to invoke cloture. If the product of the House vote is one bill, then they are not. If it is two bills, then they are. If they are not so entitled and they proceed with reconciliation anyway, then the bill will not have been validly enacted, and it will not be law.