Specialization maps for Grothendieck groups of geometric triangulated categories

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. For a field $L$, let $K_0(\mathrm{GT}_L)$ and $K_0(\mathrm{sGT}_L)$ denote the Grothendieck groups of geometric and strictly geometric triangulated categories, respectively, with relations coming from semi-orthogonal decompositions. Specialization-map conjecture. There are natural maps

$$\rho_{\mathrm{gt}}:K_0(\mathrm{GT}_K)\longrightarrow K_0(\mathrm{GT}_k)$$

and

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

The strictly geometric map is intended to be compatible with the specialization map for varieties through the natural surjection from the Grothendieck group of varieties. Its existence is the main proposed specialization statement and is not proved in general.