Biran's Pell-equation conjecture for Seshadri-like constants

Let $N>9$ be square-free, and let $d/m$ be the fundamental solution of the Pell equation $d^2-Nm^2=1$. Write $(d;m^{\times N})$ for the vector with first entry $d$ and $N$ further entries equal to $m$, and let ${\cal R}_N({\cal O}_{{\mathbf C}P^2}(1))$ denote the associated Seshadri-like constant. Biran's conjecture. Both of the following hold:

1. The vector $(d;m^{\times N})$ is nef.

$${\cal R}_N({\cal O}_{{\mathbf C}P^2}(1))\leq \frac{1}{d^2}.$$

The conjecture links continued-fraction solutions of Pell's equation with nef classes and asymptotic estimates for Seshadri-like constants on rational surfaces. The paper verifies it in many cases by recursively applying its ample-divisor construction, but the general square-free case remains open.