Equal Sum Grid Partition II

Determine whether a single straight horizontal or vertical cut can split an m×n grid into two non-empty parts whose sums are equal, or can be made equal by discounting at most one cell from either part—with the extra requirement that the part from which a cell is removed must remain connected. The core challenge is to efficiently check, for every possible cut, if the sum difference is zero or can be matched by removing a single cell whose removal does not disconnect its side (up to 1e5 total cells).