Filtration conjecture for étale homotopy sheaves

Assume that the characteristic exponent $p$ is inverted and take the étale topology. For every smooth $k$-variety $X$, write $h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$ for its zeroth étale homotopy sheaf with transfers, and let $\operatorname{Cor}(X,Y)$ denote the group of finite correspondences from $X$ to $Y$. A filtration is called strict for a morphism when the induced filtration is obtained by intersection with the target filtration. Filtration conjecture. For every smooth $k$-variety $X$, there exists a filtration

$$F^{i+1}h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)\subset F^i h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$$

with the following properties: (A) $F^0h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)=h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$ and $F^nh_0^{\text{\scriptsize{\rm {\'e}t}}}(X)=0$ for $n\geq\dim(X)+1$; (B) every correspondence $\gamma\in\operatorname{Cor}(X,Y)$ induces a morphism of homotopy sheaves compatible with the filtrations; (C) if $U$ is a dense open subvariety of $X$, then $h_0^{\text{\scriptsize{\rm {\'e}t}}}(U)\to h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$ is strict; and (D) for $n\geq0$, the quotient $F^0h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)/F^{n+1}h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$ is $n$-generated. The conjecture is intended to provide the filtration needed to extend results on $0$- and $1$-motivic sheaves to higher $n$.