The prime-power sharpness conjecture for k(m)

Let $k(m)$ be the maximum number of distinct values of $g_m(n)$ attained at a fixed endpoint in the paper's construction, and let $d(m)$ denote the number of positive divisors of $m$. Prime-power sharpness conjecture. For every prime power $m$, the bound $k(m)\leq d(m)-1$ is attained: $k(m)=d(m)-1$. The paper notes sharpness for several examples and for prime $m$, but leaves the assertion for all prime powers open.