Poisson distribution conjecture for six-packs of consecutive pth powers

For each prime $p$, let $f_p$ be the polynomial defined in the paper, and let $\rho(m)$ be the relative density of primes $p$ for which $f_p$ has exactly $m$ roots; equivalently, there are exactly $m$ pairs of consecutive $p$th powers modulo $p^2$. Six-pack distribution conjecture. For every $k\ge0$,

$$\rho(6k+2)=\frac{1}{2e^{-1/6}k!6^k}\quad\text{and}\quad\rho(6k+4)=\frac{1}{2e^{-1/6}k!6^k},$$

while $\rho(m)=0$ for all $m\not\equiv2,4\pmod 6$. In particular, the relative densities of $\tilde{\mathcal P}_{cons}$ and ${\mathcal P}_{cons}$ are respectively $1-e^{-1/6}\approx0.153518$ and $1-\frac12e^{-1/6}\approx0.576759$. This conjecture formalizes the heuristic that the number of nontrivial six-packs follows a Poisson distribution with parameter $1/6$, together with the distribution of primes modulo $6$; it is not proved in the paper.