The logarithmic Picard group conjecture for toric degenerations

Let $f:\mathcal{X}\rightarrow S$ be a toric degeneration, and write $X=\mathcal{X}_0$. Let $Z$ be the singular set of the degeneration, and let $\mathcal{M}_X$ denote the induced log structure on $X$. Logarithmic Picard group conjecture. If

$$H^1(X,\mathcal{O}_X)=H^2(X,\mathcal{O}_X)=0,$$

then

$$\operatorname{Pic}(\mathcal{X}_{\bar\eta}\setminus Z)\cong H^1(X\setminus Z,\mathcal{M}_X^{\operatorname{gp}}).$$

This conjecture identifies the Picard group of the general fibre away from the singular locus with the first cohomology of the groupification of the induced logarithmic structure on the central fibre. The supplied text presents it as a conjecture for toric degenerations, without stating a resolution or a proof.