Extension of the projective-prime dessin conjecture

Let $p$) be a projective prime, meaning that

$$p=1+q+\cdots+q^{n-1}$$

for a prime power $q$ and an integer $n\ge 2$. A dessin of type $(3,2,p)$ and degree $p$ has monodromy group ${\rm PSL}_n(q)$ when that group acts naturally on points or hyperplanes of the projective space.

Projective-prime dessin conjecture. For each projective prime $p=1+q+\cdots+q^{n-1}>3$, there exists at least one dessin of type $(3,2,p)$ and degree $p$ with monodromy group ${\rm PSL}_n(q)$.

The statement extends the proved result for Mersenne primes and is also known in the cases $n=2$ and $n=3$ described earlier in the paper; the general case remains open.