Squarefreeness conjecture for sigma-2,2 pairs

Let $p$ and $q$ be a $\sigma_{2,2}$ pair, meaning that they are distinct odd primes satisfying the paper's definition of a $\sigma_{2,2}$ pair. Squarefreeness conjecture. The integers

$$p^2+p+1$$

and

$$q^2+q+1$$

are squarefree. This is presented as a tentative open question concerning the arithmetic structure of $\sigma_{2,2}$ pairs; no resolution is given in the source.