Factorization conjecture for the derived specialization map

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $\mathrm{M}_K$ be the abelian group freely generated by isomorphism classes of smooth proper schemes over $K$, and let $K_0(\mathrm{sGT}_K)$ be the Grothendieck group of strictly geometric triangulated categories over $K$. For a smooth proper $K$-scheme with an snc model, define the candidate map $\rho:\mathrm{M}_K\to K_0(\mathrm{sGT}_k)$ by the alternating sum of the classes of the derived categories of the projective bundles over the strata. Factorization conjecture for $\rho$. The homomorphism $\rho$ factors through the canonical surjective homomorphism

$$\mathrm{M}_K\twoheadrightarrow K_0(\mathrm{sGT}_K),$$

so that there is an induced map $K_0(\mathrm{sGT}_K)\to K_0(\mathrm{sGT}_k)$ making the displayed triangle commute. This factorization is the concrete criterion that would construct the strictly geometric specialization map; the source presents it as an equivalent formulation of the conjectural map.