The self-conjugate consecutive-core and symmetric generalized Dyck path conjecture

Let $s$ and $k$ be positive integers. A self-conjugate $(s,s+1,\dots,s+k)$-core is a self-conjugate partition that is simultaneously an $s$-core, $(s+1)$-core, and so on through an $(s+k)$-core. A symmetric $(s,k)$-generalized Dyck path is a generalized Dyck path with parameters $(s,k)$ that is invariant under the relevant reflection symmetry. Self-conjugate consecutive-core conjecture. The number of self-conjugate $(s,s+1,\dots,s+k)$-cores is equal to the number of symmetric $(s,k)$-generalized Dyck paths. This extends the proved case of three consecutive core parameters established earlier in the paper; the conjecture proposes the corresponding equality for every pair of positive integers $s$ and $k$.