After reading the paper you mentioned, it seems that the reduction stated in the paper is incomplete, therefore the result may not be correct.
In the paper the author gave a reduction from 3SAT to a problem called planar max-cycle-covering (although it is not consistent with the usual sense of a covering in graph theory, it is more like max-edge-disjoint-cycles problem), and proving that the max-cycle-covering problem and max-cut problem are dual to each other on planar graphs. While the second implication is correct, the reduction part mimics a proof to the NP-hardness of planar Hamiltonian cycle problem, and there may be a flaw in it.
In that proof one need a gadget for XOR-in-series (See Fig. 3 in the Hamiltonian paper); although the construction works in the Hamiltonian ones, I see no evidence that a similar construction is possible for cycle-covering, and there are no explanations in the paper. (And we need such a gadget in the NP-hardness proof of cycle-covering, see Fig. 6 in the max-cut paper.) The main problem is that for Hamiltonian cycle we can always be sure that the middle edge C must be passed though, hence we can "transfer" the XOR by the presence of the middle edge; but in cycle-covering we cannot guarantee that edge C must be passed though unless it is within a cycle passing though a heavy-weighted edge, and this is not the case here. (In fact we cannot enforce the edge being passed though, since in Fig. 6 if an edge C is passed though then the corresponding literal in the clause is true.)
My guessing is that planar max-cycle-covering cannot simulate 3-fan-in OR function, which is required in 3SAT. Supported with the answer by @turkistany, one should believe that planar max-cut problem is polynomial time solvable.