The mixed Hodge structure conjecture for $L_2$ cohomology of Mixed Hodge Modules

Let $X$ be a complex projective manifold, let $\mathbb{M}$ be a Mixed Hodge Module on $X$, and let $H^k_2(\widetilde{X},\mathbb{M})$ denote the associated $L_2$ cohomology, equipped with its real structure, weight filtration $W$, and algebraically defined Hodge filtration $F$. Write $\overline{H^k_2}(\widetilde{X},\mathbb{M})$ for reduced cohomology. The mixed Hodge structure conjecture for $L_2$ cohomology. After taking reduced cohomology and closure of $W,F$, these data are the constituents of a functorial graded polarisable Mixed Hodge structure in the abelian category $\mathbb{R}E_f(\Gamma)$. Its mixed Hodge numbers, namely the dimensions of its $I^{p,q}$, obey the same restrictions as in the compact case. If $\mathbb{M}$ is pure polarized, $L$ is cup product by a Hodge class, and $S$ is the polarization obtained from a Saito polarization and $L_2$-Poincaré–Verdier duality, then

$$\left(\bigoplus_k \overline{H^k_2}(\widetilde{X},\mathbb{M}),L,S\right)$$

is a polarized Hodge–Lefschetz structure in $\mathbb{R}E_f(\Gamma)$. The conjecture would provide mixed Hodge and Hodge–Lefschetz structures for $L_2$ cohomology of Mixed Hodge Modules; the source states no resolution.