Vanishing of the moving Seshadri constant on the effective-cone boundary

Let $X$ be the variety under consideration, let $D\in \operatorname{Div}(X)\otimes \mathbb{Q}$ be a divisor class on the boundary of the effective cone, and let $\eta\in X$ be a general point. Boundary vanishing conjecture. One has

$$\epsilon_\eta(D)=0.$$

This conjecture asks whether the Seshadri constant vanishes at a general point for every divisor class on the boundary of the effective cone. It is posed as a difficult and important question; no resolution is given in the source.