Uniqueness of algebraic counting by mixed Hodge structures

Let $c_{alg}$ be an algebraic counting invariant of complex algebraic varieties, and let $c_{top}$ denote topological counting. The Hodge--Deligne polynomial is

$$\chi_{u,v}(X):= \sum_{i,p,q\geq 0}(-1)^i(-1)^{p+q}\operatorname{dim}_{\mathbb C}\left(Gr_F^pGr^W_{p+q}H_c^i(X,\mathbb C)\right)u^pv^q.$$

Mixed Hodge structure conjecture. The algebraic counting $c_{alg}$ specializes to $c_{top}$, and there are no further algebraic counting invariants specializing to the Hodge--Deligne polynomial $\chi_{u,v}$; equivalently, no extra structures beyond Deligne's mixed Hodge structure contribute to the algebraic counting of complex algebraic varieties. The claim is presented as a negative answer to the question of whether additional such countings exist.