Simultaneous core partitions have been widely studied since Anderson's work on the enumeration of $(s,t)$-core partitions. Amdeberhan and Leven showed that the number of $(s,s+1, \ldots, s+k)$-core partitions is equal to the number of $(s, k)$-Dyck paths. In this paper, we prove that self-conjugate $(s,s+1, \ldots, s+k)$-core partitions are equinumerous with symmetric $(s, k)$-Dyck paths, confirming a conjecture posed by Cho, Huh and Sohn.