The complementarity conditions conjecture for infinite matroid-rooted digraphs

Let $D$ be an infinite matroid-rooted digraph with source set $I$, terminal vertex $t$, and an $(I,t)$-linkage $\mathcal{P}$. For a vertex set $X\ni t$, write $\mathcal{S}(X)$ for the set of sources whose roots lie in $X$, and let $\mathsf{in}_D(X)$ denote the set of edges entering $X$. The complementarity conditions for $\mathcal{P}=\{P_i\}_{i\in I}$ and $X$ are that

$$I_{in}:=I\cap \mathcal{S}(X)\text{ is a base of }\mathcal{S}(X),$$

that the paths $\{P_i\}_{i\in I_{in}}$ lie in $D[X]$, that for $i\in I_{out}:=I\setminus \mathcal{S}(X)$ one has $|A(P_i)\cap \mathsf{in}_D(X)|=1$, and that

$$\bigcup_{i\in I_{out}}A(P_i)\supseteq \mathsf{in}_D(X).$$

Complementarity conditions conjecture. There always exist an $(I,t)$-linkage $\mathcal{P}$ and a vertex set $X\ni t$ such that $\mathcal{P}$ and $X$ satisfy the complementarity conditions.