Conjecture on optimal quantizers for the uniform distribution on a square

Let the unit square be equipped with the uniform probability distribution, and let an optimal set of $n$-means have associated quantizer points. Conjecture on optimal quantizers for the square. (i) For $n=3$, the points are symmetrically distributed with respect to one of the two diagonals or one of the two lines that bisect opposite sides of the square. (ii) For $n=4$, two points lie on a line of symmetry of the square, while the other two lie on opposite sides of that line and are equidistant from it. (iii) For $n=5$, one point is at the center $\left(\frac12,\frac12\right)$ of the square, and the other four either lie on the lines $x_1=\frac12$ and $x_2=\frac12$, equidistant from the center, or lie on the two diagonals, equidistant from the center. The conjecture is motivated by the four lines of maximum symmetry of the uniformly distributed square: its two diagonals and the two lines bisecting opposite sides. It describes the expected symmetry of optimal quantizers for three, four, and five means.