Finiteness conjecture for tau-isospectrality of two lens spaces

Let $L=L(49;1,6,15)$ and $L'=L(49;1,6,20)$ be the given lens spaces, and let $\tau$ be an irreducible representation of $\operatorname{SO}(5)$. Finiteness conjecture. There are only finitely many irreducible representations $\tau$ of $\operatorname{SO}(5)$ such that $L$ and $L'$ are $\tau$-isospectral. These spaces are $p$-isospectral for every $p$, but examples show that they are not $\tau$-isospectral for many representations; the claimed finiteness remains a question in this specific setting.