Invariant-form criterion for generic diameter-three tridiagonal pairs

Let $\Phi$ be a generic tridiagonal-pair parameter array with diameter $d=3$, acting through $A,A^*$ on $V$, and let $w_1,w_2,w_3$ be its weights. Invariant-form conjecture. The following are equivalent: (i) there exists a nondegenerate symmetric bilinear form $\langle\,\ \rangle:V\times V\to\mathbb K$ such that

$$\langle Au,v\rangle=\langle u,Av\rangle,\qquad \langle A^*u,v\rangle=\langle u,A^*v\rangle$$

for all $u,v\in V$; (ii) the weights $w_1,w_2,w_3$ are distinct. The author explicitly says the claim is uncertain, and no resolution is supplied.