The only solution which on my mind is to use tokens, but I hope, we have another solution.
Actually, tokens aren't a solution, because when the visitor hits the Thank You page, they can still be anonymous (the association of the underlying cookie from anonymous → known doesn't happen immediately but later; even if it's one second later that's too late).
Don't know why you wouldn't use tokens if you could, though. What would be wrong with tokens, if they were guaranteed to be available?
In any case, this logic requires JavaScript. You would calculate the appropriate show/hide logic on the LP (not the TY) since it has direct access to the just-submitted values. Then append the show/hide info to the URL of the Thank You page (i.e. in the query string). On the Thank You page, transform those query params into CSS classes. CSS handles the show/hide (display: none;).
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