The scalar-limit conjecture for radial Toeplitz operators

Let $n$ be the fixed positive integer defining the polyanalytic Bargmann–Segal–Fock setting, let $\mathcal{T}_n(\operatorname{RBC})$ be the radial Toeplitz C*-algebra, let $\Phi_n$ be the representation into the matrix-sequence algebra $\mathfrak{M}_n$, and let

$$\mathfrak{C}_n=\left\{A\in\mathfrak{M}_n\colon\exists v\in\mathbb{C},\ \lim_{d\to+\infty}A_d=vI_n\right\}.$$

Scalar-limit conjecture.

$$\Phi_n\bigl(\mathcal{T}_n(\operatorname{RBC})\bigr)=\mathfrak{C}_n.$$

The preceding results establish the inclusion of the represented radial Toeplitz algebra in $\mathfrak{C}_n$; the conjecture asserts that every matrix sequence with a scalar limit occurs, and the supplied parser evidence marks it as resolved.