The eventual 8g conjecture for purely non-free actions

Let $\mu(g)$ be the maximum order of a purely non-free (gpnf) action on a closed orientable surface of genus $g$. For even $g\geq 2$, the paper proves that either $\mu(g)=8g$ or

$$\mu(g)=\frac{8n}{n-4}(g-1)$$

for some integer $n>4$, and that the latter case occurs for only finitely many $g$ for each fixed $n$.

Eventual 8g conjecture.

$$\mu(g)=8g$$

for all but finitely many $g$. The authors additionally suggest that this may hold for all $g$. The preceding results leave open whether infinitely many values of $n$ can produce the exceptional form; this conjecture asserts that such exceptions occur only finitely often overall.