ELIXIR — the six-cell route
Trace ELIXIR through 1 at row 5, column 2; 2 at row 5, column 3; 3 at row 5, column 4; 4 at row 5, column 5; 5 at row 4, column 5; 6 at row 3, column 5. Reading those cells in sequence gives E → L → I → X → I → R, so both instances of I fall into their required positions.
The movement is right three times and then up twice. That produces two straight segments with a single turn, spanning three rows and four columns. All six cells touch a board edge, and one of them is a corner. Both the starting and ending cells have two open orthogonal neighbors in the raw board.
Once this route is assigned, 6 of the 21 open cells are used and 15 remain. Its long horizontal run followed by the upward finish makes the geometry straightforward to audit: confirm the three consecutive rightward steps first, then make the one turn and complete the two-cell climb.