Homological torsion growth in non-normal chains of graphs of free groups

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index non-normal subgroups $G\ge G_1\ge G_2\ge \cdots$ with exponential homological torsion growth. More specifically, we prove that simultaneously for every prime $p$, ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}\frac{\log \mathrm{\mid }{\mathrm{T}\mathrm{o}\mathrm{r}}_p(G_n^{\mathrm{a}\mathrm{b}})\mathrm{\mid }}{[G:G_n]}>0$ (where ${\mathrm{T}\mathrm{o}\mathrm{r}}_p(G_n^{\mathrm{a}\mathrm{b}})=\{g\in G_n^{\mathrm{a}\mathrm{b}}\mathrm{\mid }g\text{ has order a power of }p\}$).