The largest-unitization conjecture for the Moyal plane

Let $\mathcal{A}_1=\mathcal{D}_{L^2}(\mathbb{R}^{2N})$ be the algebra of Moyal multipliers acting on the relevant Hilbert space, let $\mathcal{B}(\mathbb{R}^{2N})$ denote the proposed preferred unitization, and let ${D\mkern-11.5mu/\,}$ be the Dirac operator. The commutator $[{D\mkern-11.5mu/\,},a]$ is required to be bounded for every $a$ in the unitization. Largest-unitization conjecture. The preferred unitization is the largest Moyal multiplier algebra satisfying this bounded-commutator condition:

$$\widetilde{\mathcal{A}} = \mathcal{B}(\mathbb{R}^{2N})$$

and this algebra is the largest Moyal multiplier algebra of $\mathcal{A}_1$ such that $[{D\mkern-11.5mu/\,},a]$ is bounded for each $a\in\widetilde{\mathcal{A}}$. The conjecture concerns whether the orientation and boundedness conditions uniquely determine the preferred compactification; it is strengthened by the result attributed to Melo and collaborators, but the supplied text does not establish the full maximality assertion.