The Nomura–Terwilliger conjecture on the Terwilliger algebra corner

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let $A,A^*$ be a tridiagonal pair on $V$. 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 $\mathcal{T}$ be the $\mathbb{K}$-subalgebra of $\operatorname{End}(V)$ generated by $\mathcal{D}$ and $\mathcal{D}^*$.

Nomura–Terwilliger conjecture. With respect to this notation, the following hold: (i) $E^*_0\mathcal{T}E^*_0$ is generated by $E^*_0\mathcal{D}E^*_0$; and (ii) the elements of $E^*_0\mathcal{D}E^*_0$ mutually commute.

This is a stronger and more detailed form of the conjecture that $E^*_0\mathcal{T}E^*_0$ is commutative. The source gives no resolution status.