Generalized Ambro conjecture for weak log Fano pairs

Let $(Y,B)$ be an $\epsilon$-klt weak log Fano pair of dimension $n$ with $Y$ having at worst terminal singularities. For a $\mathbb{Q}$-Cartier divisor $G$ satisfying $G+B\geq0$, define

$${\rm glct}(Y,B;G)=\sup\{t\in[0,1]\cap\mathbb{Q}\mid (Y,B+tG)\text{ is lc}\}.$$

Generalized Ambro conjecture. Fix $0<\epsilon<1$ and an integer $n>0$. Then there exists a number $\mu(n,\epsilon)>0$ depending only on $n$ and $\epsilon$ such that

$$\inf\{ {\rm glct}(Y,B;G)\mid G\sim_\mathbb{Q}-(K_Y+B),\ G+B\geq0\}\geq\mu(n,\epsilon).$$

This stronger formulation allows $G$ to be non-effective, subject to $G+B\geq0$, and is needed for the paper's argument. The supplied text gives no resolution status, so it is recorded as open with low confidence.