Additional hyperplanes conjecture for directed-cycle model singular loci

Let $n \geq 3$, and let $\mathcal{M} = (G, \textit{In}, \textit{Out}, \textit{Leak})$ be an $n$-compartment directed-cycle model with $\textit{In}=\{1\}$ and $\textit{Out}=\{p\}$, where $1 \leq p \leq n$. Assume that $\mathcal{M}$ is generically locally identifiable, equivalently, that it is leak-interlacing. For $a \in ([p-1] \smallsetminus \textit{Leak})$ and $\ell \in (\textit{Leak} \cap [p])$, define the corresponding rate parameters $k_{a+1,a}$, $k_{\ell+1,\ell}$, and leak parameter $k_{0\ell}$. Additional hyperplanes conjecture. For every such $a$ and $\ell$, the singular locus of $\mathcal{M}$ contains the hyperplane

$$\{k_{\ell+1,\ell}+k_{0\ell}=k_{a+1,a}\}.$$

This conjecture predicts further components of the singular locus beyond those established for the case $\textit{In}=\textit{Out}=\{1\}$. Proving it would extend the known description of singular loci for generically locally identifiable, or leak-interlacing, directed-cycle compartment models.