The common-unitary normal-form conjecture for GLT algebras

Let $\mathscr E$ be the space of matrix sequences, let $\mathscr M_D$ be the measurable complex-valued functions on $D$ modulo almost-everywhere equality, and let $\mathfrak C_D$ denote the class of algebras of matrix sequences with symbols in $\mathscr M_D$. Suppose that an algebra $\mathfrak A\subseteq\mathfrak C_D$ is induced by a subalgebra $\mathscr A\subseteq\mathscr E$ and an algebra homomorphism $s:\mathscr A\to\mathscr M_D$. Let $\mathscr S_U$ denote the class of sequences admitting the corresponding normal form through a unitary sequence $\{U_n\}_n$.

Common-unitary normal-form conjecture. Any algebra $\mathfrak A\subseteq\mathfrak C_D$ induced by such $\mathscr A$ and $s$ admits a unitary sequence $\{U_n\}_n$ such that $\mathscr A\subseteq\mathscr S_U$. Equivalently, every sequence in the algebra admits a normal form through the same unitary base change.

The claim asks for simultaneous normal forms for all sequences in an algebra, extending the normal-form result for GLT sequences. No proof or resolution is supplied in the paper.