The involutive antiautomorphism conjecture for tridiagonal pairs

Assume $\mathbb F$ is algebraically closed. Let $V$ be a vector space over $\mathbb F$ with finite positive dimension, and let $A,A^*$ be a tridiagonal pair on $V$. The involutive antiautomorphism conjecture. There exists a unique antiautomorphism $\dagger$ of $\operatorname{End}(V)$ that fixes both $A$ and $A^*$, and

$$X^{\dagger\dagger}=X$$

for every $X\in\operatorname{End}(V)$.

Such an antiautomorphism would provide a canonical involutive symmetry of the endomorphism algebra associated with the pair. The claim is listed among the paper's directions for extending structural results beyond Krawtchouk type.