Matrix-valued spherical-function product expansion conjecture

Let $l>0$, let $n$ be a nonnegative integer, and let $\Phi(w,t)$ be the matrix-valued spherical function introduced in section $3$. For $i\leq j$, consider products of these functions and matrices $A_k$.

$$\Phi(i,t)\Phi(j,t)=\sum_{k=\min\{j-i-l,0\}}^{j+i+l}A_k\Phi(k,t).$$

Matrix-valued product-expansion conjecture. The product admits a unique expansion of this form, with matrix coefficients $A_k$, for all nonnegative $n$.

The claim extends the usual scalar angular-momentum range when $l=0$ by allowing extra terms at both ends. The source says it is well known for $l=0$ and $n=0$, while the asserted general case is presented as a conjecture.