Cho–Huh–Sohn conjecture for self-conjugate consecutive-core partitions

For positive integers $s$ and $k$, a partition is a self-conjugate $(s,s+1,\ldots,s+k)$-core partition if it is equal to its conjugate and is simultaneously an $s$-core, an $(s+1)$-core, through an $(s+k)$-core. A ballot $(s,k)$-path is a lattice path from $(0,0)$ to $(s,n)$ using up steps $U_k=(k/2,1)$, down steps $D_k=(k/2,-1)$, and horizontal steps $H_\ell=(\ell,0)$ for $1\leq\ell<k$, never lying below the $x$-axis; an $(s,k)$-Dyck path is a ballot $(s,k)$-path of height $0$, and it is symmetric if reflection about the line $x=s/2$ leaves it unchanged. Cho–Huh–Sohn conjecture. For given positive integers $s$ and $k$, the number of self-conjugate $(s,s+1,\ldots,s+k)$-core partitions is equal to the number of symmetric $(s,k)$-Dyck paths. The paper presents this as a conjecture and states that it confirms it, so the conjecture is solved.