Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it always possible to find a collection of arc disjoint, flow-carrying $s$-$t$-paths that cover all arcs $a$ with flow $x_a=T$?

This question came up in work on the scheduling of arc outages to maximize the total flow over a time horizon.

** Update: ** To avoid Brendan's counterexample let's assume that every node can be reached from the source $s$, and from every node the sink $t$ can be reached. Fixing a flow $x$ we can reduce the capacity of every arc to its flow value (and delete zero capacity arcs) without really changing the problem. Now the question is as follows:

Given a network with the property that for every node $v\in N\setminus\{s,t\}$ the sum of the capacities of the incoming arcs equals the sum of the capacities of the outgoing arcs, can the arcs of maximum capacity be covered by a collection of arc-disjoint $s$-$t$-paths?

In this formulation it seems possible that someone has looked at this problem before.