Invertibility conjecture for a cyclotomic circulant matrix

Let $q=p^s$ be a prime power, let $Q=q^m$, and let $N$ be a divisor of $Q-1$. Write $N_1=(Q-1)/N$ and let $A$ be the circulant matrix defined in equation from the relevant Gaussian periods. Invertibility conjecture. The circulant matrix $A$ is invertible. This conjecture is motivated by numerical computations suggesting that the associated polynomial does not vanish at the relevant roots of unity; its validity would support the characterization of constant $b$-symbol weight irreducible cyclic codes and is noted as a question of interest in cyclotomy.