You said that something has to be detectable in order to be defined as existing. We can't detect quarks, so by your definition of existence, quarks don't exist.
You're quibbling over the definition of detection. Since everything we know about the world outside of our brains is mediated by intervening stages of detection via sensors, from our physical senses to machines like telescopes we have built to enhance those senses, to the constructions our brains make from these raw inputs to create an experience which we naively take to be direct input from the world, to the schemes of understanding we construct on top of that to interpret our various inputs, it's all inferred from interpreting the effects of the "things". You can't touch an atom either, but do you dismiss a scanning electron microscope as being detection of the atoms? You said we can only detect them by inferring their existence by observing their effects. I say that's exactly the same thing we do with every other thing we speak of as existing.
I'd say, if we have a model of reality, and everything we detect fits into that model, and there's nothing missing from the model that would need to be there to explain anything, or any projection from that model, then we have existence wrapped up.
Provisionally, until a gap presents itself. Then we need to adjust the model, if and when required. Trying to squeeze a "maybe gap" into the model where there just isn't one, as in the case of ghosts for instance, is not a genuine challenge to the model. (Because there's no requirement for an extra force beyond the ones we know already in order to account for everything we currently detect). If ghosts occurred, we could detect them.
For all intents and purposes, if something "exists" beyond the model, or beyond the universe, if it doesn't impinge on this model or this universe, it really doesn't exist!
This is maybe a subtle point: Navigator has already misconstrued me to mean that things don't exist until we discover them, which is obviously nonsensical. I'm talking here in the ideal of the perfectly congruent case of our completed model matching exactly the reality of the universe. If the universe does not contain it, then it can not be said to exist. If our model is congruent with the universe, then we can detect it, via it's effects… because the only way we can detect anything is via its effects.