Odd-colour phase-transition conjecture for multicoloured connectivity

Let $m(n,r,s,k)$ denote the largest order guaranteed for a $k$-connected subgraph using at most $s$ colours in every $r$-colouring of $K_n$. Let $s\geq 3$, and let $n$ and $k$ be sufficiently large. Set $r=2s-1$.

Odd-colour phase-transition conjecture. One has

$$m(n,2s-1,s,k)\leq n-\left(\frac{2s(r-2)}{r^2-2r+2}\right)k =n-\left(1+\frac{2s-5}{4(s-1)^2+1}\right)k,$$

where $r=2s-1$.

This conjecture concerns the value to which the function jumps when the number of colours is odd, just below the threshold $r=2s$. The case $s=2$ is known, but the authors conjecture that the analogous general formula gives an upper bound for all $s\geq 3$.