A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups

For positive integers $q$ and $m$ with $m\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answe