Lovász–Cherkassky theorem without the countability hypothesis

Let $G$ be a graph and let $T\subseteq V(G)$ such that there is no $X\subseteq V(G)\setminus T$ for which $d_G(X)$ is an odd natural number. A system $\mathcal{P}$ of edge-disjoint $T$-paths has the required cut property when, for every $t\in T$, one can choose exactly one edge from each path having $t$ as an end-vertex so that the resulting edge set $C$ is a cut separating $t$ from $T-t$.

Lovász–Cherkassky conjecture. Under these hypotheses, there exists a system $\mathcal{P}$ of edge-disjoint $T$-paths having the required cut property for every $t\in T$.

The paper proves the corresponding statement when $T$ is countable and conjectures that countability can be omitted. The unrestricted case is presented as open.