Cover-by-elementary-operation conjecture for positroids

Let $\sigma$ and $\pi$ be positroids on $[n]$ of ranks $k-1$ and $k$, respectively. For $A\subset[k]$, let $\overleftarrow{\rho_A}(\pi)$ denote the positroid obtained from $\pi$ by the indicated reverse operation. Cover-by-elementary-operation conjecture. Then

$$\sigma\lessdot\pi\quad\Longleftrightarrow\quad\sigma=\overleftarrow{\rho_A}(\pi)\text{ for some }A\subset[k].$$

This conjecture summarizes the paper's findings and is supported by evidence generated using SageMath; its general validity remains open.