Conjecture on the exact value of m(n,r,1,k)

Let $m(n,r,1,k)$ denote the largest integer such that every $r$-colouring of the edges of $K_n$ contains a monochromatic $k$-connected subgraph on at least $m(n,r,1,k)$ vertices. Let $n,k,r\in\mathbb{N}$ satisfy

$$r\geqslant 3,\qquad n\geqslant 2r(k-1)+1,$$

with $r-1$ a prime power and $n-r(k-1)$ divisible by $(r-1)^2$.

Exact-value conjecture. Under these conditions,

$$m(n,r,1,k)=\frac{n-k+1}{r-1}.$$

The preceding upper and lower bounds motivate this exact formula. The lower bound on $n$ is sharp in the stated sense, since $m(n,r,1,k)=0$ when $n\leqslant 2r(k-1)$. The conjecture identifies the remaining value in the specified arithmetic range.