Asymptotic strict mixed Hodge inequality for powers of a space

Let $X$ be the space under consideration, let $MH_{X^n}(t,u,v)$ and $MH^{\pi}_{X^n}(t,u,v)$ denote respectively the mixed Hodge polynomial and homotopical mixed Hodge polynomial of its $n$-fold product $X^n$, and let $(\mathbb R_{>0})^3$ be the positive octant. Let $\varepsilon$ be a positive real number such that $0<\varepsilon\ll1$.

Asymptotic strict mixed Hodge inequality conjecture. There exists a positive integer $n_0$ such that, for every $n\geqq n_0$,

$$MH^{\pi}_{X^n}(t,u,v)<MH_{X^n}(t,u,v)$$

for every $(t,u,v)\in[\varepsilon,\infty)^3\subset(\mathbb R_{>0})^3$. The preceding results establish analogous inequalities on compact positive cubes, while the conjecture asks for one uniform bound over the entire unbounded region $[\varepsilon,\infty)^3$.