Mordell's Pellian equation conjecture

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, let $\mathcal{O}_K$ be the ring of algebraic integers of $K$, and let $\varepsilon>1$ be the fundamental unit of $\mathcal{O}_K$. Set

$$\omega=\begin{cases}\sqrt{d}&\text{if }d\equiv 2,3\pmod 4,\frac{1+\sqrt{d}}{2}&\text{if }d\equiv 1\pmod 4.\end{cases}$$

Write $\varepsilon=x+y\omega$ with $x,y\in\mathbb{N}_0$. Mordell's Pellian equation conjecture. If $d\in\mathbb{P}$ and $d\equiv 3\pmod 4$, then $d\nmid y$. The paper's abstract states that recently discovered counterexamples to Mordell's Pellian equation conjecture are discussed, so its status is refuted.