Minimal quotient complexity characterizes quotients of the shift algebra

Let $\mathsf{A}$ be a finite-dimensional, indecomposable, unital commutative operator algebra of minimal quotient complexity. Here, minimal quotient complexity means that every $r$-dimensional quotient has a completely isometric representation by $r\times r$ matrices. Minimal-complexity characterization conjecture. If $\mathsf{A}$ has dimension $r$, then there exists a closed, finite-codimensional ideal $\mathsf{I}\subset\mathsf{Shift}_{r-1}$ with $\operatorname{sa}(\mathsf{I})=\{0\}$ such that

$$\mathsf{A}\cong\mathsf{Shift}_{r-1}/\mathsf{I}\quad\text{or}\quad\mathsf{A}\cong(\mathsf{Shift}_{r-1}/\mathsf{I})^t.$$

Moreover, if $\mathsf{A}\cong\mathsf{A}^t$, then $\mathsf{A}$ is a quotient of $\mathsf{Shift}_1$. This conjecture would extend the result for trivial unitizations, which are automatically indecomposable, to all finite-dimensional indecomposable algebras of minimal quotient complexity; its status is left unresolved in the source.