3 problems

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.

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

Bispectral algebra conjecture. The algebra of all such $\Theta(x)$ is the algebra of polynomials