Existence of a strong phase transition for the p-adic Ising-Vannimenus model

Let $p\geq3$ and let $a,b\in\mathcal{E}_p$ with $b\neq1$. Let $\Delta$ be the set of all solutions of the system referred to as. If $(k,p)=1$ and

$$\frac{p-1}{(k,p-1)}$$

is even, then

$$\Delta\cap\bigl((\mathbb{Q}_p\setminus\mathbb{Z}_p)\times(-\mathcal{E}_p)\times(-\mathcal{E}_p)\bigr)\neq\emptyset.$$

Strong phase-transition conjecture. Under these assumptions, the displayed intersection is nonempty.

The preceding results prove the existence of a phase transition, but the existence of a strong phase transition remains open in the paper and is asserted conditionally on this conjecture.