The Markov degree conjecture for the incidence configuration of the loopless complete graph K5

Let $A$ be the incidence matrix of the complete graph $K_5$ without self-loops, namely

$$A=\begin{pmatrix} 1&1&1&1&0&0&0&0&0&0\\ 1&0&0&0&1&1&1&0&0&0\\ 0&1&0&0&1&0&0&1&1&0\\ 0&0&1&0&0&1&0&1&0&1\\ 0&0&0&1&0&0&1&0&1&1 \end{pmatrix}.$$

Let ${\mathbb N}A$ denote the semigroup generated by the columns of $A$, and for each $\Bb\in {\mathbb N}A$ let $A_\Bb$ be the corresponding fiber configuration. Markov degree conjecture.

$$\max_{\Bb\in {\mathbb N}A} \operatorname{MD}(A_\Bb)=2.$$

This asserts that every fiber configuration arising from this incidence configuration has Markov degree at most two, despite the fact that the matrix itself has Markov complexity at least six and Graver complexity fifteen according to the preceding computation. The claim is presented as a conjecture, and no resolution is given in the source.