A Ramsey Type problem for highly connected subgraphs

Bollob\'{a}s and Gy\'{a}rf\'{a}s conjectured that for any $k, n \in \mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices leads to a $k$-connected monochromatic subgraph with at least $n-2k+2$ vertices. We find a counterexample with $n = \lfloor 5k-2.5-\sqrt{8k-\frac{31}{4}} \rfloor$, thus disproving the conjecture, and we show the conclusion holds for $n > 5k-2.5-\sqrt{8k-\frac{31}{4}}$ when $k \ge 16$.