The coefficient-of-$q^{2s-2}$ conjecture for the Fibonacci polynomials $F_s(q)$

Let $F_s(q)$ be the recursively defined polynomials

$$F_0(q)=1,\qquad F_1(q)=1,\qquad F_s(q)=(1+q+2q^2)F_{s-1}(q)+q^3F_{s-2}(q)\quad(s\ge2).$$

The coefficient-of-$q^{2s-2}$ conjecture. For all $s\ge2$, the coefficient of $q^{2s-2}$ in $F_s(q)$ is the number of compositions of $s+9$ into $s$ parts, none of which is $2$ or $3$, namely OEIS sequence A134465. The claim was verified computationally for $s\le10$ and remains open as stated.