The Seshadri criterion for the strict interior of the nef cone

Let $X$ be a projective variety and let $\alpha\in\operatorname{Nef}^k(X)$. Suppose that there exists $\varepsilon>0$ such that

$$\operatorname{\varepsilon\,}(\alpha;x)\geq\varepsilon$$

for all $x\in X$. Seshadri criterion. Then $\alpha$ is in the strict interior of $\operatorname{Nef}^k(X)$. This would characterize the strict interior of the nef cone by a uniform positive lower bound for the Seshadri function. The source states the assertion as a conjectural criterion, and no resolution is supplied.