Recently Lau-Jeyaseeli-Shiu-Arumugam introduced the concept of the "Sudoku colourings" of graphs -- partial $χ(G)$-colourings of $G$ that have a unique extension to a proper $χ(G)$-colouring of all the vertices. They introduced the Sudoku number of a graph as the minimal number of coloured vertices in a Sudoku colouring. They conjectured that a connected graph has Sudoku number $n-1$ if, and only if, it is complete. In this note we prove that this is true.