Infinite Mader theorem for internally vertex-disjoint T-paths

Let $G=(V,E)$ be a possibly infinite graph and let $T\subseteq V$. An internally vertex-disjoint $T$-path system is accompanied by $X\subseteq V\setminus T$ and a partition $\mathcal{Y}$ of $V\setminus(T\cup X)$. For each $Y\in\mathcal{Y}$, define

$$B_Y=\\{v\in Y: v\text{ has a neighbour in }V\setminus(X\cup Y)\\}.$$

After deleting $X$ and all edges inside the parts $Y$, no $T$-path remains; the paths cover $X$ and all but at most one vertex of each $B_Y$. Each path either meets $X$ once and each $B_Y$ at most once, or meets $X$ not at all, meets one uniquely determined $B_Y$ twice, and every other $B_Y$ at most once; moreover, at most one path meets each $B_Y$ exactly once.

Mader's internally-disjoint-path conjecture. Such a system $\mathcal{P}$, set $X$, and partition $\mathcal{Y}$ exist.

This is proposed as an infinite complementary-slackness analogue of Mader's finite minimax theorem for internally vertex-disjoint $T$-paths. No resolution is given in the supplied text.