Large-column existence conjecture

Prove that $(r\times c,v)$-near triple arrays exist for every $r\geq 3$, $c\geq r(r-1)$, and $v\geq c$.

Background

The paper proves that near triple arrays with three rows exist whenever $c\geq 6=3(3-1)$ and $v\geq c$. It also proves a non-existence result for each fixed $r\geq 4$ at $c=r(r-1)-1$ and $v=c+2$, showing that the threshold in the conjecture is motivated by a sharp obstruction.

The conjecture proposes that the bound $c\geq r(r-1)$ is sufficient for existence for all numbers of symbols at least equal to the number of columns.