The torsion polynomial conjecture for lens spaces

Let $L(p,q)$ be a lens space with $p$ prime and $p\ge3$, and let $\tau(p,q)$ denote its associated torsion invariant. The homeomorphism classification identifies parameters precisely along the orbits $\{\pm q^{\pm1}\}$.

Torsion-polynomial conjecture. For every prime $p\ge3$, the map $q\mapsto \tau(p,q)$ is constant precisely on the orbits $\{\pm q^{\pm1}\}$. Equivalently, $\tau(p,q)$ distinguishes the homeomorphism classes of lens spaces $L(p,q)$.

The invariant is already known to depend only on the homeomorphism-class orbit, and computations up to $p\le 400$ support the converse. Whether it separates all such orbits in general remains open.