The generating-set conjecture for Leonard systems

Let $\Phi$ be a Leonard system with primitive idempotents $E_0,E_1,\ldots,E_d$ and dual primitive idempotents $E^*_0,E^*_1,\ldots,E^*_d$ in its associated algebra $\mathcal A$. The generating-set conjecture. For every integer $r$ with $0\leq r\leq d$, the elements

$$E^*_0,E^*_1,\ldots,E^*_r,E_r,E_{r+1},\ldots,E_d$$

together generate $\mathcal A$. This is a structural generation statement for Leonard systems. The surrounding source discusses a related span problem and notes that that problem is solved for Leonard pairs, but gives no resolution of this generating assertion.