The simsun and zigzag refinements of the zigzag numbers

For $n\geq 1$, define $B(n,k)$ and $C(n,k)$ by

$$I_n(-1,t)=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}B(n,k)t^{n-1-k}$$

and

$$\widetilde{I}_n(-1,t)=\sum_{k=0}^{n-2}C(n,k)t^{k+1}.$$

Here a simsun permutation is a permutation with the stated simsun property, and a big return is the stated statistic on an alternating permutation. Simsun and zigzag refinement conjecture. The coefficient $B(n,k)$ is the number of simsun permutations in $S_{n-1}$ with $k$ descents, while $C(n,k)$ is the number of alternating permutations in $S_n$ with $k$ big returns. These identities would give two distinct natural refinements of the zigzag numbers $E_n$, since $I_n(-1)=E_n$. They are supported experimentally by the displayed small cases; the source does not provide a resolution.