The equality conjecture for the homotopy-exponent valuation bound

Let $p$ be any prime, let $\alpha,l\in\mathbb{N}$, and let $n,r\in\mathbb{Z}$ with $n\ge 2p^{\alpha}-1$. The theorem gives a lower bound for the $p$-adic valuation of the relevant quantity $R$ in terms of $\tau_p(\{r\}_{p^{\alpha}},\{n-r\}_{p^{\alpha}})$. Equality conjecture. Equality in Theorem is attained whenever $l\ge\lfloor n/p^{\alpha}\rfloor$ and

$$l\equiv\left\lfloor\frac{r}{p^{\alpha}}\right\rfloor+\left\lfloor\frac{n-r}{p^{\alpha}}\right\rfloor\ \left(\operatorname{mod}\ (p-1)p^{\lfloor\log_p(n/p^{\alpha})\rfloor}\right).$$

The claim is based on extensive Maple calculations; the source provides no proof or resolution, so its status remains open.