Odd order C₄-face-magic projective grid graphs

For a graph G = (V, E) embedded in the projective plane, let F(G) denote the set of faces of G. Then, G is called a Cₙ-face-magic projective graph if there exists a bijection f: V(G) → {1, 2, …, |V(G)|} such that for any F ∈ F(G) with F ≅ Cₙ, the sum of all the vertex labels around Cₙ is a constant S. We consider the m × n grid graph, denoted by P m,n , embedded in the projective plane in the natural way. Let m ≥ 3 and n ≥ 3 be odd integers. It is known that the C₄-face-magic value of a C₄-face-