Mutation-equivalence conjecture for canonical TP-bases of positroid cells

Let a positroid cell in the nonnegative Grassmannian be given. Its canonical TP-basis is the set of Plücker variables labeling the dots of the associated $\text{ \vbox{\hrule width 0.35em height 0.04 em} \vbox{\offinterlineskip \text{\kern -0.02em\vrule height 0.65em width 0.04em \hspace{0.1em}} } }$-diagram. Two TP-bases are mutation equivalent if one can be obtained from the other by a sequence of Plücker mutations induced by 3-term Plücker relations.

Canonical TP-basis conjecture. For any positroid cell, the canonical TP-basis can be transformed by Plücker mutations into a TP-basis consisting of variables coming from unique paths.

The conjecture has been checked for all cells in $Gr_{k,n}^{\mathrm{tnn}}$ with $n\leq 8$ and proved for weakly-connected cells. The general case, including non-weakly-connected cells, remains open in the source.