The classification conjecture for sharp tridiagonal systems

Let $d$ be a nonnegative integer, and let

$$(\{\theta_i\}_{i=0}^d;\{\theta_i^*\}_{i=0}^d;\{\zeta_i\}_{i=0}^d)$$

be scalars in $\mathbb{K}$. A sharp tridiagonal system is a tridiagonal system whose eigenspaces have the relevant one-dimensional sharpness property, and its parameter array is the displayed sequence. For $0\leq i\leq d$, define

$$\eta_i(\lambda)=(\lambda-\theta_d)(\lambda-\theta_{d-1})\cdots(\lambda-\theta_{d-i+1}),$$ $$\eta_i^*(\lambda)=(\lambda-\theta_d^*)(\lambda-\theta_{d-1}^*)\cdots(\lambda-\theta_{d-i+1}^*).$$

Classification conjecture. There exists a sharp tridiagonal system $\Phi$ over $\mathbb{K}$ with this parameter array if and only if the following conditions hold: (i) $\theta_i\neq\theta_j$ and $\theta_i^*\neq\theta_j^*$ whenever $i\neq j$; (ii) 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 $2\leq i\leq d-1$; and (iii) $\zeta_0=1$, $\zeta_d\neq0$, and

$$0\neq\sum_{i=0}^d\eta_{d-i}(\theta_0)\eta_{d-i}^*(\theta_0^*)\zeta_i.$$

When these conditions hold, $\Phi$ is unique up to isomorphism of tridiagonal systems.

This is the proposed parameter-array classification of sharp tridiagonal systems. The supplied text gives no evidence that it has been proved or disproved.