HIGH — the four-letter path
Trace HIGH through these four cells: 1 at row 3, column 5; 2 at row 4, column 5; 3 at row 5, column 5; 4 at row 5, column 4. The letters appear as H → I → G → H, so both H cells have distinct positions in this spelling.
Begin by moving down twice, then turn left for the final letter. The route has one turn and two straight segments, spanning three rows and two columns. All four cells lie on a board edge, and the finish has two open orthogonal neighbors. Once this word is assigned, 18 open cells remain for the other three entries.
Use the repeated H as a route check rather than assuming the first H you notice must begin the word. The starting H has three open orthogonal neighbors, but the correct trace commits to two downward steps before leaving that straight run. If your fourth letter does not follow a left turn, the sequence has departed from this path.