Relative weight-detection conjectures for étale homology

Let $X\in G'$ and let $r,n\ge 0$. For each Zariski point $x$ of $X$, let $i_x$ be the corresponding embedding, and let $H$ be the product of the theories $H^{et}_{{\mathbb{Q}_l}}(x)\circ i_x^!$. Let $M\in\operatorname{Obj}DM_c(X)$, and let $E_2^{pq}T_{{w_{Chow}}(X)}(H,M)$ be the terms of the associated weight spectral sequence. If a function $\delta'$ is defined, let $H^{et}_{{\mathbb{Q}_l}}(X)$ denote the relative perverse étale homology theory.

Relative weight-detection conjectures. The conjecture $WD^n(X)$ holds if

$$E_2^{pq}T_{{w_{Chow}}(X)}(H,M)=0$$

for all $q\in\mathbb{Z}$ and $p>n$ implies $M\in DM_c(X)_{{w_{Chow}}\ge -n}$. Its restricted version $WD_r^n(X)$ requires additionally that $M\in\widetilde{N}_r(X)$. If $\delta'$ is defined, the conjecture $WD'^n(X)$ holds if

$$E_2^{pq}T_{{w_{Chow}}(X)}(H^{et}_{{\mathbb{Q}_l}}(X),M)=0$$

for all $q\in\mathbb{Z}$ and $p>n$ implies $M\in DM_c(X)_{{w_{Chow}}\ge -n}$. Its restricted version requires additionally that $M\in\widetilde{N}'_r(X)$, defined using $\delta'$.

These conjectures seek to detect relative Chow-weight bounds using étale homology, either pointwise or through a relative perverse realization. The source presents the assertions as conjectures and supplies no evidence of resolution.