Let $\bar{X}$ be a complete smooth variety over $\mathbb{C}$ and $D$ be a simple normal crossing divisor. Denote $X:=\bar{X}\backslash D$. Then it is known that $H^\ast(X,\mathbb{C})$ admits a canonical mixed Hodge structure.

Denote $j:X\to\bar{X}$ be the open immersion and $A_X^\ast$ be the complex of sheaves of differential forms on $X$. Let $F^p$ be the subcomplex of $A_X^\ast$ consists of $(i,j)$-forms such that $i\geq p$. My question is

Dose $(j_\ast A_X^\ast,j_\ast F^\bullet,\tau^{\leq\bullet} j_\ast A_X^\ast)$ induces the canonical mixed Hodge structure on $H^\ast(X)\simeq H^\ast(\bar{X},j_\ast A_X^\ast)$?