The numerical triviality conjecture for log motives

Let $S$ be an fs log scheme and let $M,N$ be objects of the category $\mathrm{LM}(S)$ of log motives. For a morphism $f:M\to N$, say that $f$ is numerically equivalent to zero if $\operatorname{Tr}(gf)=0$ for every morphism $g:N\to M$; write $f\sim g$ when $f-g$ is numerically equivalent to zero. Let $\mathrm{LM}_{\mathrm{num}}(S)$ denote the category modulo this numerical equivalence.

Numerical triviality conjecture. In $\mathrm{LM}(S)$, $f\sim g$ implies $f=g$; equivalently,

$$\mathrm{LM}(S)=\mathrm{LM}_{\mathrm{num}}(S).$$

This asserts that numerical equivalence agrees with the homological equivalence built into the category of log motives. The source provides no resolution status.