Explicit orthonormal basis conjecture for $SU(2,2)$ coherent states

Let $n>1$, let $F_{2,2,n}^{U(n)}$ be the symmetric subspace, and for $\ell,m,r,s\in\mathbb{N}$ define

$$a_{r,s}^{\ell,m}:=(a_r^n a_s^n-a_{r-1}^n a_{s-1}^n)a_{\ell}^{n+r+s}a_m^{n+r+s},\qquad a_k^n:=\tbinom{n+k-1}{k}.$$

For variables $Z_{11},Z_{12},Z_{21},Z_{22}$, define

$$\psi_{r,s}^{\ell,m}:=\frac{1}{\ell!r!s!m!\sqrt{a_{r,s}^{\ell,m}}}\sum_{i=0}^{\min(r,s)}(-1)^i\frac{\tbinom{r}{i}\tbinom{s}{i}}{\tbinom{n+r+s-2}{i}}Z_{11}^{\ell+i}Z_{12}^{r-i}Z_{21}^{s-i}Z_{22}^{m+i}.$$

$SU(2,2)$ coherent-state basis conjecture. The set $\{\psi_{r,s}^{\ell,m}:\ell,m,r,s\in\mathbb{N}\}$ forms an orthonormal basis of $F_{2,2,n}^{U(n)}$. The formulas have been checked only for small values of the parameters and arbitrary $n>1$. The conjecture would provide the explicit orthonormal basis not supplied in the general treatment of $SU(p,q)$ coherent states; proving orthonormality is described as challenging because fixed-total-degree subspaces are generally degenerate.