Existence of a measurement realizing finite quantum relative entropy

Let $\omega_1$ and $\omega_2$ be states, let $S(\omega_1\Vertomega_2)$ denote their quantum relative entropy, and let $\mathrm{M}(\omega_1\precomega_2)$ be the set of measurement schemes used to obtain the corresponding central measures. Existence conjecture. If

$$S(\omega_1\Vertomega_2)<\infty,$$

then

$$\mathrm{M}(\omega_1\precomega_2)\neq\emptyset.$$

This would provide a statistical measurement framework for examining quantum relative entropy whenever it is finite. The source does not state any resolution, so the conjecture remains open.