Optimal regularity conjecture for K-finite matrix coefficients on compact symmetric spaces

Let $G$ be a semisimple Lie group with finite center, let $U$ be the compact simply connected semisimple Lie group with complexified Lie algebra corresponding to the compact dual, and let $K\subset U$ be the associated maximal compact subgroup. Write

$$r=\lfloor \kappa(G)\rfloor,\qquad \delta=\kappa(G)-r,$$

and let $U_r$ denote the subset of regular points. A matrix coefficient is $K$-finite if it is finite under the left and right actions of $K$.

Optimal regularity conjecture. Any $K$-finite matrix coefficient of a unitary representation of $U$ is in $C^{(r,\delta)}(U_r)$, and this regularity is optimal.

This conjecture concerns the sharp Hölder regularity of coefficients on the regular set of the compact symmetric space $U/K$. The paper proves the corresponding result for $K$-bi-invariant coefficients and establishes the full statement in special cases, while the general $K$-finite assertion remains open.