Ito–Terwilliger endpoint algebra conjecture for tridiagonal systems

Let $\mathbb{K}$ be a field, let $V$ be a finite-dimensional vector space, and let $\Phi=(A;\{E_i\}_{i=0}^d;A^*;\{E_i^*\}_{i=0}^d)$ be a tridiagonal system. Let $\mathcal{D}$ and $\mathcal{D}^*$ be the $\mathbb{K}$-subalgebras of $\operatorname{End}(V)$ generated by $A$ and $A^*$, respectively, and let $T$ be the subalgebra generated by both. Let $E_0^*$ be the first primitive idempotent of $A^*$. Ito–Terwilliger's endpoint algebra conjecture. The following hold: (i) $E_0^*TE_0^*$ is a field with identity $E_0^*$; (ii), on viewing $\mathbb{K}E_0^*$ as a field with identity $E_0^*$, the field $E_0^*TE_0^*$ is an $r$-dimensional field extension of $\mathbb{K}E_0^*$, where $r=\dim E_0^*V$. This conjecture concerns the algebra generated at an endpoint idempotent; the supplied text gives no resolution status.