Conjecture on the rightmost entries of the new Catalan triangle

Let the rows of the Catalan triangle given by the proposition be numbered starting from row $1$. The rightmost entries in rows are considered according to their indices modulo $3$.

Rightmost-entry conjecture. The rightmost numbers in rows numbered $2+3i$, for $i=0,1,\ldots$, are the Catalan numbers, and the rightmost numbers in rows numbered $1+3i$, for $i=0,1,\ldots$, are the central elements of the $(1,2)$-Pascal triangle.

The observed sequences begin $1,1,2,5,14$ and $1,3,9,30,105$, respectively. The conjecture was verified for permutations of length at most $24$; its general validity is not established here.