Let $X$ be a smooth, complex surface of general type such that $\Omega_X$ is globally generated, and let $n \geq 2$ be a natural number.

Question. Is there a way to compute $h^i(X, \, \operatorname{Sym}^n \Omega_X)$, where $i \in \{0, \, 1\}$, in terms of the invariants of $X$ (Chern numbers, geometric genus, irregularity, etc.)? Or, at least, is it possible to compute it under some additional assumptions (either on $X$ or on $n$)?

Remark. I know how to compute $\chi(X, \, \operatorname{Sym}^n \Omega_X)$, see MO397682.