Derived invariance of Hodge numbers

Let $X$ and $Y$ be nice varieties over the complex numbers, meaning smooth, projective, geometrically irreducible schemes of finite type, and suppose they are derived equivalent through a complex-linear exact equivalence $D^b(X)\simeq D^b(Y)$. Let $h^{p,q}(X)$ and $h^{p,q}(Y)$ denote their Hodge numbers.

Derived Hodge-invariance conjecture. Derived equivalent nice varieties defined over the complex numbers have the same Hodge numbers.

This is predicted by the relationship between Chow motives and cohomological realizations, and the paper studies implications from the conjectural zeta-function invariance to this Hodge-theoretic statement. It is presented as an open conjectural consequence.