The structural formula conjecture for cohomology characters

Fix $n$ and $p$. For each $m\geq1$ and $0\leq i\leq n-3$, let $P_i(m)\in A$ be a character of degree $2m-2+n-i$, with $P_0(m)=\mathrm{Prim}(m)$. Let $F^r$ denote the $r$-fold Frobenius transform and $s_\lambda$ the Schur function.

Structural formula conjecture. For $d\geq e>0$,

$$\kappa(d,e)=\sum_{r,m,i}F^rP_i(m)\cdot s_{e-p^rm,\underbrace{0,\ldots,0}_{n-2},d+p^r(2-m-n+i)+n-1},$$

where the sum ranges over triples $(r,m,i)$ for which the displayed Schur-function arguments are weakly decreasing.

This conjecture proposes the general form of all computed cohomology characters, but the source states that the complexity of the problem prevents an exact formula and supplies no proof.