Al-Salam–Carlitz moment Pfaffian conjecture

Let $U_n^{(a)}(x;q)$ be the Al-Salam–Carlitz polynomials, and let $G_n(a;q)=\sum_{k=0}^n\left[{{n}\atop{k}}\right]_q a^k$ be their moment sequence, where $\left[{{n}\atop{k}}\right]_q=\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}$. For $n\geq1$, set $G_{-1}(a;q)=0$. Al-Salam–Carlitz moment Pfaffian conjecture. The identities

$$\mathop{\rm Pf}\biggl((q^{i-1}-q^{j-1})G_{i+j-3}(a;q)\biggr)_{1\leq i,j\leq 2n}=a^{n(n-1)}q^{\frac13\lfloor n/2\rfloor(16\lfloor n/2\rfloor^2-1)-(-1)^n4\lfloor n/2\rfloor^2-2\lfloor n/2\rfloor\cdot\lfloor (n-1)/2\rfloor}\prod_{k=1}^{n}(q;q)_{2k-1},$$

and

$$\mathop{\rm Pf}\biggl((q^{i-1}-q^{j-1})G_{i+j-2}(a;q)\biggr)_{1\leq i,j\leq 2n}=a^{n(n-1)}q^{\frac13\lfloor n/2\rfloor(16\lfloor n/2\rfloor^2-1)-(-1)^n4\lfloor n/2\rfloor^2}\prod_{k=1}^{n}(q;q)_{2k-1}\sum_{k=0}^{n}q^{\lfloor (n-2k)^2/2\rfloor}\left[{{n}\atop{k}}\right]_{q^2}a^k$$

would hold, where $\lfloor x\rfloor$ is the greatest integer not exceeding $x$. These are experimentally supported Pfaffian evaluations for a $q$-moment sequence; their status is left open.