On the maximal dimension of an irreducible representation of the symmetric group

We prove that the maximal dimension $d_N$ of an irreducible representation of the symmetric group $S_N$ satisfies $$d_N=\sqrt{N!}{e}^{-(\mathfrak{d}+o(1))\sqrt{N}},N\to \mathrm{\infty },$$ for some constant $\mathfrak{d}>0$. This answers a question raised by Vershik–Kerov in 1985.