Saito's conjecture on cohomological characteristic classes

Let $k$ be a perfect field, let $X$ be a closed subscheme of a smooth scheme over $k$, and let $\Lambda$ be the finite local coefficient ring in the setup. Let $\mathcal F$ be a constructible complex of $\Lambda$-modules of finite tor-dimension on $X$. Write $cc_{X,0}(\mathcal F)\in {\rm CH}_0(X)$ for the characteristic class defined via the characteristic cycle, and set $cc_X(\mathcal F):=cc_{X,0}(\mathcal F)$. The cohomological characteristic class is $C_{X/k}(\mathcal F)\in H^0(X,\mathcal K_{X/k})$, where $\mathcal K_{X/k}=Rh^!\Lambda$ for the structural morphism $h:X\to {\rm Spec}k$. Saito's conjecture. One has

$$C_{X/k}(\mathcal F)={\rm cl}(cc_X(\mathcal F))\quad\text{in}\quad H^0(X,\mathcal K_{X/k}),$$

where ${\rm cl}:{\rm CH}_0(X)\to H^0(X,\mathcal K_{X/k})$ is the cycle class map. This conjecture asserts that the cohomological characteristic class can be computed from the characteristic cycle; the paper confirms it in the quasi-projective case, while the general case is the broader assertion attributed to Saito.