Highly connected subgraphs with large chromatic number

For integers $k\ge 1$ and $m\ge 2$, let $g(k,m)$ be the least integer $n\ge 1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. Refining the recent result Gir~ao and Narayanan that $g(k-1,k)\le 7k+1$ for all $k\ge 2$, we prove that $g(k,m)\le \max (m+2k-2,\lceil (3+\frac{1}{16})k\rceil )$ for all $k\ge 1$ and $m\ge 2$. This sharpens earlier results of Alon, Kleitman, Saks, Seymour, and Thomassen, of Chudnovsky, Penev, Scott, and Trotignon, and of Penev, Thomass'{e}, and Trotignon. Our result implies that $g(k,k+1)\le \lceil (3+\frac{1}{16})k\rceil$ for all $k\ge 1$, making a step closer towards a conjecture of Thomassen from 1983 that $g(k,k+1)\le 3k+1$, which was originally a result with a false proof and was the starting point of this research area.