Pell-equation lower-bound conjecture for Seshadri constants on Picard-rank-one surfaces

Let $X$ be a smooth projective surface with Picard number $1$, and let $L$ be the ample generator of the Néron–Severi space, with $N=L^2$. Let $(\ell_0,k_0)$ denote the primitive solution of Pell's equation

$$\ell^2-Nk^2=1.$$

Pell-equation lower-bound conjecture. One should have

$$\varepsilon(L;1)\geqslant \left\{ \begin{array}{ccl} \sqrt{N} & \text{ if } & N\,\text{ is a square}\\ \frac{Nk_0}{\ell_0} & \text{ if } & N\,\text{ is not a square} \end{array}\right..$$

This conjecture predicts a stronger numerical restriction on the one-point Seshadri constant for surfaces of Picard number one than the general bound established earlier in the paper; the source provides no resolution of the conjecture.