Universal-cover conjecture for operator-algebraic strict deformation quantization

Let $(M,\omega)$ be a symplectic manifold, and let $C^{\infty}_c(M)$ denote the compactly supported smooth functions on $M$. A strict deformation quantization is understood in the sense of a continuous family of $C^*$-algebras whose classical member is $C^{\infty}_c(M)$ with pointwise multiplication and whose commutator has the prescribed Poisson-bracket limit. Universal-cover conjecture. There is a strict deformation quantization of $C^{\infty}_c(M)$ whose underlying $C^*$-algebra is a subalgebra of bounded linear operators on $L^2(M,\omega)$ if and only if the universal cover of $M$ is $T^*\mathbb{R}^n$. This proposes a geometric characterization of when a symplectic manifold admits the specified operator-algebraic strict deformation quantization. The source gives no evidence of a resolution.