The three-term recurrence conjecture for matrix-valued spherical functions

For a given nonnegative integer $\ell$, let $\Phi(t,p)$ be the matrix whose rows are the vectors $H(t)$ corresponding to $k=0,1,2,\dots,\ell$. The matrices $A_p$, $B_p$, and $C_p$ are required to be independent of $t$.

Three-term recurrence conjecture. There exist matrices $A_p$, $B_p$, and $C_p$, independent of $t$, such that

$$A_p\Phi(t,p-1)+B_p\Phi(t,p)+C_p\Phi(t,p+1)=\frac{t^2+1}{t}\Phi(t,p).$$

This is the proposed bispectral three-term recursion in the spectral parameter $p$ for the matrix-valued spherical functions. The supplied text does not state whether the conjecture has been proved or disproved.