The manifold inheritance conjecture for isospectral lens orbifolds

Let $L$ and $L'$ be $(2n-1)$-dimensional lens orbifolds. A lens orbifold has a manifold structure when its defining parameters are coprime to its group order; in that case it is a lens space. For $0<p\leq n-1$, say that $L$ and $L'$ are $p$-isospectral when their spectra on $p$-forms agree.

The manifold inheritance conjecture. If $L$ and $L'$ are $p$-isospectral for some $0<p\leq n-1$ and $L$ is a lens space, then $L'$ is also a lens space.

The corresponding statement for $p=0$ is given as a theorem in the paper, while the general orbifold setting admits counterexamples for positive degrees. Computational evidence suggests that this stronger inheritance property holds for lens orbifolds.