The unilateral frame number conjecture for rigid matrix model operators

Let $H^2$ be the Hardy space on the disk, let $S$ be the unilateral shift of multiplicity $m$ on $(H^2)^m$, and let $K_\Theta\subset (H^2)^m$ be a nonzero $S$-coinvariant subspace associated to a rigid matrix $\Theta\in (H^\infty)^{m\times m}$. Using the continuous Gelfand extension of $\Theta$ to the maximal ideal space $\mathcal{M}(H^\infty)$, write $S_\Theta$ for the corresponding model operator.

The unilateral frame number conjecture.

$$\vartheta_+(S_\Theta)=\sup_{x\in\mathcal{M}(H^\infty)}\dim\operatorname{Ker}\Theta(x).$$

This conjectures a formula for the minimum number of vectors needed to generate a frame by iterating the model operator. The paper proves one inequality, reduces the other to a particular case, and gives an example supporting the formula; the conjecture is therefore left open.