Closure of the circuit reduction property under gluing

A matrix $A$ has the circuit reduction property if, for every minimal Markov basis $M$ of $A$, distance reduction for the circuits of $A$ implies that $M$ is distance reducing. Denote by $\mathcal A$ the set of matrices with the circuit reduction property. A gluing of matrices $A$ and $B$ is denoted $A \circ B$.

Closure conjecture. The set $\mathcal A$ is closed under gluing: if $A,B\in\mathcal A$ admit a gluing, then

$$A\circ B\in\mathcal A.$$

The conjecture would extend circuit-based distance-reduction criteria through iterated gluings. Its validity is left open in the source.