The logarithmic Chow–Künneth decomposition conjecture

Let $X$ be a projective vertical log smooth fs log scheme of relative dimension $d$ over $S$, and let $h(X)$ be its log motive in the category $\mathrm{LM}(S)$ of log motives. Write $h^i(X)_{\ell}$ for the $\ell$-adic realization of a summand and $H^i(X)_{\ell}$ for the $i$-th $\ell$-adic cohomology sheaf.

Logarithmic Chow–Künneth conjecture. The motive $h(X)$ should admit a decomposition

$$h(X)=h^0(X)\oplus h^1(X)\oplus\dots\oplus h^{2d}(X)$$

in $\mathrm{LM}(S)$ such that $h^i(X)_{\ell}=H^i(X)_{\ell}$.

This predicts a cohomologically graded decomposition of the log motive, analogous to a Chow–Künneth decomposition. The source provides no resolution status.