  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.
