Elimination of null edge vectors on reducible PBDTP networks

Let $({\mathcal N},\mathcal O,\mathfrak l)$ be a reducible PBDTP network representing a point $[A]\in {\mathcal S}_{\mathcal M}^{\text{\tiny TNN}}\subset {\operatorname{Gr}}^{\text{\tiny TNN}}(k,n)$ in an irreducible positroid cell. Suppose it has finitely many edges $e_1,\dots,e_s$ carrying null vectors, so that $E_{e_l}=0$ for $l\in[s]$. Elimination of null vectors on reducible PBDTP networks. Using the gauge freedom for unreduced graphs, one may always change the weights on $\mathcal N$ so that the resulting network $(\widetilde{\mathcal N},\mathcal O,\mathfrak l)$ still represents $[A]$ and satisfies $\widetilde E_e\ne0$ for every edge $e\in\widetilde{\mathcal N}$. The conjecture asserts that reducible networks can be reweighted without changing the represented positroid point so as to remove all null edge vectors; this is the common content of the paper's two prose formulations and the displayed conjecture.