Infinite Gallai theorem for vertex-disjoint T-paths

Let $G=(V,E)$ be a possibly infinite graph and let $T\subseteq V$. A system $\mathcal{P}$ of vertex-disjoint $T$-paths and a set $X\subseteq V$ are required to satisfy: for every component $Y$ of $G-X$, the paths contained in $Y$ cover all but at most one vertex of $T\cap Y$, and $X\subseteq V(\mathcal{P})$ with $|V(P)\cap X|\leq 1$ for every $P\in\mathcal{P}$.

Gallai's conjecture. There exists such a system $\mathcal{P}$ and set $X$.

This is proposed as an infinite generalization of Gallai's theorem; the supplied text does not state that it has been proved.