Gimbert's cyclotomic irreducibility conjecture

For integers $i>2$ and $k>1$, define the cyclotomic polynomial $\Phi_i(x)$ and

$$F_{i,k}(x)=\Phi_i(1+x+\cdots+x^k).$$

Gimbert's cyclotomic irreducibility conjecture. If $k$ is even, then $F_{i,k}(x)$ is reducible in $\mathbb Q[x]$ if and only if $i\mid(k+2)$, and in that case it has exactly two factors. If $k$ is odd, then $F_{i,k}(x)$ is irreducible in $\mathbb Q[x]$ if and only if $i\mid 2(k+2)$, and in that case it has exactly two factors. The conjecture concerns the factorization over $\mathbb Q$ of characteristic polynomials associated with almost Moore digraphs and was attributed in the source to Gimbert; the supplied text gives no resolution status.