Conjectural formula for the reproducing kernel on the positive cone

Let $\Omega$ be a bounded symmetric domain of rank $r$, with invariants $p$, $b$, $d$, and $q_\Omega$, and let $\Gamma_\Omega$ denote the associated gamma function. Let $m$ and $\nu$ be parameters, and let $q$ be the nonnegative integer satisfying

$$q<\frac{\nu-p+1}{2}\leq q+1.$$

Assume that $m\geq rq+1$. Reproducing-kernel formula. The reproducing kernel $S^m_\nu$ on $\mathbf R_+^r$ at the origin is

S^m_\nu(x,0)=c^q_\nu\,{}_2\mathcal F_1\Bigg(\begin{matrix}-q,-b-\nu+2p+q-2r\\#4\end{matrix}\Bigg|-x\Bigg),

where

$$c^q_\nu=\frac{\Gamma_\Omega(\nu-p-q+2r+b)\Gamma_\Omega(q_\Omega)\Gamma_\Omega(p+q)}{\pi^d\Gamma_\Omega(\nu-p-q+2r-q_\Omega)\Gamma_\Omega(q_\Omega+q)\Gamma_\Omega(p)}.$$

The formula is presented as the outcome of extensive calculations involving Faraut–Korányi hypergeometric functions and spherical polynomials; its resolution is not specified in the source.