Specifically, let $G$ be an undirected graph and $S$ and $T$ be two (possibly intersected) subsets of vertices, is the size of the minimum edge cut for $S$ and $T$ equal to the maximum number of pairwise edge-disjoint paths from $S$ to $T$?
Here a path from $S$ to $T$ means a path from a vertex $u$ in $S$ to a vertex $v$ in $T$ where $u\neq v$, and the minimum edge cut for $S$ and $T$ is the minimum number of edges whose removal cuts off all paths from $S$ to $T$.
Note the edge-connectivity version of Menger's theorem handles the special case where $S$ and $T$ are sets of a single vertex.
Intuitively this is true, but how to prove it? I tried to prove it from the max-flow min-cut theorem, but the trick of constructing a flow network by adding vertices $s$ and $t$ as well as edges from $s$ to $S$ and from $T$ to $t$ does not work because $S$ and $T$ may be intersected.