Enumeration of canon permutations avoiding (213,312) when n = 3

Prove or determine whether c_3^k(213,312) = 4C_k for every k ≥ 1, where C_k is the kth Catalan number.

Background

This is a concrete instance of the preceding question about c_3^k(Λ) for arbitrary subsets Λ of S_3. The conjectured count is four times the kth Catalan number and concerns canon permutations avoiding the two patterns 213 and 312.

The statement is presented as an experimentally suggested formula rather than a proved result.