Divisibility conjecture for seven-dimensional isospectral lens orbifolds

Let $L(q;s)$ and $L(q;s')$ be $7$-dimensional lens orbifolds with fundamental group of order $q$. They are non-isometric if they are not isometric, and $\{p\}$-isospectral when their spectra on $p$-forms agree.

The divisibility conjecture. If $L(q;s)$ and $L(q;s')$ form a pair of non-isometric $\{2\}$-isospectral $7$-dimensional lens orbifolds, then $3$ divides $q$; respectively, if they form a pair of non-isometric $\{3\}$-isospectral $7$-dimensional lens orbifolds, then $5$ divides $q$.

The claim is motivated by computational evidence reported in the source. It gives necessary divisibility conditions on the fundamental-group order, but no proof or resolution is supplied.