Ito–Terwilliger classification conjecture for sharp tridiagonal systems

Let $\mathbb{F}$ be a field, let $d$ be a nonnegative integer, and let $(\{\theta_i\}_{i=0}^d;\{\theta_i^*\}_{i=0}^d;\{\zeta_i\}_{i=0}^d)$ be a sequence of scalars in $\mathbb{F}$, with $\tau_i,\eta_i,\tau_i^*,\eta_i^*$ the associated polynomials. A sharp tridiagonal system is a tridiagonal system whose parameter array is this sequence. Ito–Terwilliger's classification conjecture. There exists a sharp tridiagonal system $\Phi$ over $\mathbb{F}$ with this parameter array if and only if: (i) $\theta_i\ne\theta_j$ and $\theta_i^*\ne\theta_j^*$ whenever $i\ne j$; (ii) $\zeta_0=1$, $\zeta_d\ne0$, and

$$\sum_{i=0}^d\eta_{d-i}(\theta_0)\eta^*_{d-i}(\theta_0^*)\zeta_i\ne0;$$

(iii) the expressions

$$\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i},\qquad \frac{\theta^*_{i-2}-\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i}$$

are equal and independent of $i$ for $2\le i\le d-1$. If these conditions hold, then $\Phi$ is unique up to isomorphism of tridiagonal systems. This is the proposed classification of sharp tridiagonal systems by their parameter arrays; the paper proves that the μ-conjecture implies it and records verification in small diameter, but the general assertion remains open.