Local stability conjecture for R-Cartier boundaries

Let $\Gamma\subseteq[0,1]$ be a finite set and let $X\ni x$ be a germ. Local stability conjecture. There exists a positive real number $\tau$ depending only on $\Gamma$ and $X\ni x$ such that, whenever $B'\leq B$, $\lVert B-B'\rVert<\tau$, $B\in\Gamma$, and $K_X+B'$ is $\mathbb{R}$-Cartier, then $K_X+B$ is $\mathbb{R}$-Cartier. This asks for uniform local stability of the $\mathbb{R}$-Cartier property under small coefficient increases; no resolution status is supplied in the provided text.