Conjectural sums of minimal permutation representation invariants for groups of orders $p^4$ and $p^5$

For a finite group $G$, let $4\Delta(G)4$ denote the invariant used in the paper, and let the sums range over isomorphism classes of groups of the indicated order. Numerical-sum conjecture. For every prime $p>3$,

$$\sum_{|G|=p^{4}}\Delta(G)=1+\frac{5}{p}+\frac{11}{p^{2}}+\frac{9}{p^{3}},$$

and

$$\sum_{|G|=p^{5}}\Delta(G)=1+\frac{7}{p}+\frac{34+2\gcd(p-1,3)+\gcd(p-1,4)}{p^{2}}+\frac{54}{p^{3}}+\frac{24}{p^{4}}.$$

These formulas extend the explicitly computed sums for groups of orders $p$, $p^2$, and $p^3$; the source presents the cases $p^4$ and $p^5$ as conjectural numerical results, with no resolution supplied here.