Distance-irreducible elements for three-variable complete intersections

Let

$$A=\begin{pmatrix}a_1&a_2&a_3\end{pmatrix}$$

be a complete intersection with a unique minimal Markov basis

$$M=\{b:=(b_1,-b_2,0),\ c:=(c_1,c_2,-c_3)\},$$

and suppose that $b_1>b_2$. Let $D(A)$ denote the set of distance-irreducible elements of $A$.

Distance-irreducible basis conjecture. If $M$ is not distance reducing, then

$$D(A)=M\quad\Longleftrightarrow\quad 2c_1<b_1+b_2.$$

This conjecture predicts exactly when the distance-irreducible elements equal the unique minimal Markov basis in this complete-intersection setting. The source gives an example showing that a unique minimal Markov basis need not equal $D(A)$; the general assertion remains open.