The algebraic cohomology exactness conjecture for resolutions

Let $X$ be a singular projective variety, let $\phi:\widetilde{X}\to X$ be any resolution of singularities, and let $E$ be the exceptional divisor. Let $p$ be an integer such that

$$\operatorname{dim}(X_{\mathrm{sing}})\le p-1.$$

The cohomology sequence

$$H^{2p}(X)\to H^{2p}(\widetilde{X})\to H^{2p}(E)$$

is exact. The algebraic cohomology exactness conjecture. Taking algebraic cohomology groups preserves this exactness; namely,

$$H^{2p}_A(X)\to H^{2p}_A(\widetilde{X})\to H^{2p}_A(E)$$

is exact. This conjecture asks whether exactness survives after restricting to algebraic cohomology classes; the source does not specify whether it has been resolved.