Bahturin–Regev invariance conjecture for minimal quantum commutative decompositions

Let $R=(\mathscr{R},\theta)$ be a regular quantum commutative algebra of quantum length $m$. Its regular quantum commutative decomposition is minimal if no two columns, equivalently no two rows, with different numbers in the quantum decomposition matrix $M^{R}$ are equal.

Bahturin–Regev conjecture. The following statements hold:

1. $\mathscr{R}$ is minimal if and only if

$$\det M^{R}\neq 0.$$

2. If $\mathscr{R}$ is minimal, then the quantum length $m$ and the determinant of the corresponding matrix are invariants, independent of the particular minimal decomposition.

These assertions characterize minimality through the decomposition matrix and establish that the resulting quantum length and determinant do not depend on the chosen minimal decomposition. The supplied source does not state whether the conjecture has been resolved.