The power-of-2-or-3 conjecture for rigid-type partial geometries

Let $\mathcal{S}$ be a partial geometry $pg(s,t,\alpha)$ admitting an abelian Singer group such that every line has trivial stabilizer in the group; call $\mathcal{S}$ of rigid type. Let $v$ denote the number of points of $\mathcal{S}$.

Power-of-2-or-3 conjecture. If $\mathcal{S}$ is a $pg(s,t,\alpha)$ of rigid type, then

$$v=2^m\quad\text{or}\quad v=3^n$$

for some nonnegative integers $m$ and $n$.

The conjecture is motivated by the paper's computational search and nonexistence results, which leave several hypothetical rigid-type cases. The source does not report a proof or disproof, so the conjecture remains open.