Palindromicity and sign-pattern conjecture for a graded Euler characteristic

For $n\geq 0$, define

$$F_n(q)=\left(\sum_{k=0}^n(-1)^{n-k}q^{2(n-k)}S^o[n,k]\right)-1,$$

where $S^o[n,k]$ are the ordered type $A$ $q$-Stirling polynomials. Palindromicity and sign-pattern conjecture. The polynomial $F_n(q)$ is palindromic after ignoring signs, has the same number of positive and negative coefficients, and its lower-degree half of nonzero coefficients is positive while the remaining half is negative. The polynomial arises as a conjectural graded Euler characteristic related to generalized exterior derivatives on super coinvariant algebras; the source gives no resolution status.