Triangular-number conjecture for the smallest triangle containing unit disks

Let $n$ unit disks be given, and let $m$ be a positive integer such that

$$n=\frac{m(m+1)}{2}.$$

The smallest triangle containing these disks is an equilateral triangle of side length $2(m-1)+2\sqrt{3}$. Triangular-number conjecture. If $n=\frac{m(m+1)}{2}$, then the smallest triangle containing $n$ unit disks is the equilateral triangle of side $2(m-1)+2\sqrt{3}$. The assertion is proved in the paper for $m=1,2,3$, but the approach stops at $m=4$; the general case remains open and is motivated by the expected efficiency of Groemer packings.