Platonic-case non-asphericity conjecture for relative presentations

Let $\mathcal{Q}$ be the relative presentation considered in the paper, with parameters $l>0$, $k\neq 0$ and non-trivial elements $g,h\in G$. Define

$$\mu=\frac{1}{|g|}+\frac{1}{|h|}+\frac{1}{|gh^{-1}|}.$$

Say that Case $(P)$ holds when $\mu>1$ and $g\neq h$. The Platonic-case conjecture. If Case $(P)$ holds, then $\mathcal{Q}$ is non-aspherical. The source records non-diagrammatic reducibility or non-weak asphericity in several parameter ranges, but expects these results to extend to non-asphericity generally; no resolution is supplied.