Maximal-weight conjecture for simple holomorphic eta quotients of odd prime-power level

For a prime $p$ and an integer $n>3$, let $f_{p,n}$ be the eta quotient defined by

$$f_{p,n}:=\left\{\begin{array}{cl} \frac{\eta_p^p\eta_{p^{n-1}}^{(p-1)^2}\prod_{s=1}^{n/2-1}\eta_{p^{2s-1}}^{p^2-3p+1}\eta_{p^{2s}}^{p^2-2p+2}}{(\eta\eta_{p^n})^{p-1}}&\text{if }n\text{ is even},\\ \frac{(\eta_p\eta_{p^{n-1}})^p\prod_{s=1}^{n-1}\eta_{p^s}^{p^2-3p+2}}{(\eta\eta_{p^n})^{p-1}}&\text{if }n\text{ is odd and }p\ne2. \end{array}\right.$$

Here $\eta_m$ denotes $\eta(mz)$, and a simple holomorphic eta quotient is one that is both primitive and quasi-irreducible. Maximal-weight conjecture. For any integer $n>3$ and any odd prime $p$, there are no simple holomorphic eta quotients of level $p^n$ and of weight greater than that of $f_{p,n}$.

The preceding theorem establishes that $f_{p,n}$ itself is simple, so the conjecture asserts that it has maximal weight among simple holomorphic eta quotients at these odd prime-power levels. Numerical evidence motivates the claim; its general validity remains open.