Maximum flow with constraints
First a quick note, the flow entering a vertex is equal to the flow leaving. I'll just refer to it as the amount of flow through a vertex.

Second, note that we can say $a \leq b \wedge b \leq a$, thus we can add the constraint $a = b$ for two vertices.

Then we can re-use the NP-hard result from the paper that you quoted:


  "A negative disjunctive constraint states that a certain pair of arcs cannot be simultaneously used for sending flow"


The reduction is done by putting two proxy vertices in the middle of the respective arcs, let's call them $a$ and $b$.

We then also add two vertices $a'$ and $b'$, directly connecting to the source, and add the constraints $a = a'$ and $b = b'$. Finally we add vertex $c$, and connect it to $a'$, $b'$  and the sink. Note that the flow through $c = a' + b' = a + b$.

We finish the reduction by adding constraints $c \leq a$ and $c \leq b$. Now $a+b \leq a$ and $a+b \leq b$ implies that either $a=0$ or $b=0$ (or both).