Farnik–Szemberg–Szpond–Tutaj-Gasinska conjecture on Seshadri constants

Let $X$ be a surface with Picard group $\mathbb{Z}H$, where $H$ is an ample divisor and $H^2$ is not a perfect square. Let $(q,p)$ be a primitive solution of

$$p^2-H^2q^2=1.$$

For a point $x\in X$, let $\varepsilon(H,x)$ denote the Seshadri constant of $H$ at $x$, and set

$$\varepsilon(H,1)=\max_{x\in X}\varepsilon(H,x).$$

Farnik–Szemberg–Szpond–Tutaj-Gasinska conjecture. One has

$$\varepsilon(H,1)\geq\frac{q}{p}H^2.$$

This conjecture concerns lower bounds for Seshadri constants on surfaces of Picard rank one. The supplied context gives no evidence of a resolution, so its status remains open.