Triangular-row conjecture for optimal n-mean configurations

Let $n$ be a positive integer, and set

$$N=\lfloor\sqrt{2n}\rfloor,$$

and

$$J=N-\left|n-\frac{N(N+1)}{2}\right|.$$

An optimal configuration is a set of $n$ points minimizing the quantization error for the uniform distribution on an equilateral triangle. Arrange its points in $N$ rows, with the $j$th row indexed by $j$.

Triangular-row conjecture. For most $n$, there is an optimal configuration with $N$ rows. The $j$th row has $j$ points for $j\leq J$. If $n>N(N+1)/2$, the rows with $j>J$ each have one extra point, so the $j$th row has $j+1$ points; if $n<N(N+1)/2$, they each have one fewer point, so the $j$th row has $j-1$ points.

The conjecture describes the row structure suggested by the numerical search and by the observed proximity of optimal configurations to triangular lattices. It remains open in the supplied source.