Conjecture on the six-colour two-colour connectivity function

Let $n,k \in \mathbb{N}$, with $n$ sufficiently large compared to $k$.

The conjecture. The maximum order of a $k$-connected subgraph using at most two colours in every $6$-colouring of $K_n$ satisfies

$$m(n,6,2,k) = \frac{3n}{4} - O(k).$$

This is the conjectured asymptotic value for the case $r=6$; the paper notes that the corresponding value is known up to a factor of $2$ in general, while the case $r=5$ has already been determined asymptotically.