Let $X$ be a proper singular variety over $k=\overline{\mathbb F}_p,$ irreducible of dimension $d.$ Let $H^*(X)$ and $IH^*(X)$ be the $l$-adic cohomology groups and $l$-adic intersection cohomology groups of $X,$ resp. Then, is the natural map $H^*(X)\to IH^*(X)$ compatible with the cup-product on $H^*(X)$ and the intersection product on $IH^*(X)?$

**Background and Motivation:** Given $X_0/\mathbb F_q$ an $\mathbb F_q$-structure of $X,$ one deduces a Galois action on $H^*(X)$ and $IH^*(X),$ with respect to which they are "mixed" (the 2nd one being pure), and the weight filtrations $W$ on both of them are independent of the choice of $X_0.$ One has a natural morphism
$$
H^n(X) \to IH^n(X)
$$
which factors as
$$
Gr^W_n H^n(X) \hookrightarrow IH^n(X),
$$
and this turns out to be injective.

As $X$ is singular, Poincaré duality might fail, i.e. the cup-product $$ H^i(X)\otimes H^{2d-i}(X) \to H^{2d}(X), $$ which is Galois equivariant, may be degenerate. This is the case when $H^i(X)$ (resp. $H^{2d-i}(X)$) is not pure of weight $i$ (resp. $2d-i$) for the reason of Galois, and $W_{i-1}H^i(X)$ (resp. $W_{2d-i-1}H^{2d-i}(X)$) is contained in the left kernel (resp. right kernel) of the cup-product pairing. I would like to know if this is the only obstruction for Poincaré duality to hold, namely they are exactly the left/right kernel.

Since the intersection pairing
$$
IH^i(X)\otimes IH^{2d-i}(X)\to\mathbb Q_l(-d)
$$

is perfect (I don't know if the pairing has a geometric definition in char. $p$ --- $D_XIC_X\simeq IC_X$ is the only reason I know), if my question in the beginning has a positive answer, then the cup-product pairing on $Gr^W_*H^*(X)$ will be perfect.

**Correction:** The argument above for non-degeneracy on $Gr^W_*H^*(X)$ is wrong. Here's a counter-example. Let $X$ be the projective cone of a projective smooth curve of genus $g,$ either over char. 0 or $p.$ Then $H^1(X)=0$ but $H^3(X)$ is of dimension $2g$ and pure of weight 3.