Hook alternating property conjecture for matrix-valued spherical functions

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 $i<j$, write

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

Hook alternating property conjecture. The coefficients $A_k$ with $k$ in the range $j-i\leq k\leq j+i$ have the hook alternating property.

The source introduces this terminology after observing that, for $l>0$, extra terms occur at both ends of the expansion range. The supplied text does not define the hook alternating property more precisely or give evidence of its resolution.