Specialization map for strictly geometric triangulated categories

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. For a field $L$, let $K_0(\mathrm{sGT}_L)$ be the Grothendieck group generated by bounded derived categories of coherent sheaves of smooth projective varieties over $L$, modulo relations induced by semi-orthogonal decompositions. Specialization conjecture for strictly geometric triangulated categories. There is a natural group homomorphism

$$\rho_{\mathrm{sgt}}:K_0(\mathrm{sGT}_K)\longrightarrow K_0(\mathrm{sGT}_k).$$

Such a map would provide a specialization operation for derived categories modulo semi-orthogonal decompositions and would yield the weaker specialization statement discussed after the question about preserving full exceptional collections. The source proposes the map but does not establish it in general.