The authors' uniform moving-Seshadri-constant conjecture

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, and let $L$ be an ample divisor on $X$. For a divisor $D$, write $\varepsilon(\lVert D\rVert;x)$ for its moving Seshadri constant at a closed point $x$.

Uniform moving-Seshadri-constant conjecture. There exist integer-valued functions $c(n)$ and $d(n)$ and a real-valued function $e(n)$, all depending only on $n$, such that

$$\varepsilon\bigl(\bigl\lVert K_X+c(n)K_X+d(n)L \bigr\rVert;x\bigr)\geq\frac{1}{e(n)}$$

for every closed point $x\in X$.

The paper identifies this statement as sufficient for its Fujita-type jet-ampleness conjecture. It is presented as an ingredient of the authors' approach rather than as an established theorem.