Graph Theory – Is there a Ford Fulkerson run for each flownet in which all extension paths consist of forward edges only?

Is there a Ford Fulkerson algorithm for each flownet in which all extension paths consist of forward edges only?

I've seen this claim for a flow network where all edges have c (e) = 1, and I tried to find a counter example for networks with different capacities, but could not.

Is this statement always correct? and if so, do we need reverse edges just because we can not know which paths have only forward edges while we are executing the FF algorithm?

Many Thanks!