The generalized hypergeometric expression conjecture for matrix-valued spherical functions

For a given nonnegative integer $\ell$, let $H(t)$ denote the matrix-valued spherical functions constructed from the corresponding values $k=0,1,\dots,\ell$. The generalized hypergeometric function

$${}_{p+2}F_{p+1}\left(\begin{array}{c}a,b,s_1+1,\dots,s_p+1\c,s_1,s_2,\dots,s_p\end{array};z\right)$$

is a special class of generalized hypergeometric functions.

Generalized hypergeometric expression conjecture. For a given $\ell\geq 0$, the components of the functions $H(t)$ are expressible in terms of generalized hypergeometric functions of the form

$${}_{p+2}F_{p+1}\left(\begin{array}{c}a,b,s_1+1,\dots,s_p+1\c,s_1,s_2,\dots,s_p\end{array};z\right).$$

This conjecture proposes a single generalized-hypergeometric description of the matrix-valued spherical functions, extending the explicitly known expressions as linear combinations of classical hypergeometric functions. The supplied text does not state whether the conjecture has been proved or disproved.