The major-index generating-function conjectures for rectangular posets

Let $EN_{s,t}(\sigma)[q]$ and $NE_{s,t}(\sigma)[q]$ denote the major-index generating functions for the indicated pattern-avoiding linear extensions of rectangular posets. Let $CW_n$ be the set of Catalan words of length $2n$, and define

$$c_n(q)=\sum_{\pi\in CW_n}q^{\operatorname{maj}(\pi)},$$

where $\operatorname{maj}$ is the major index. The major-index generating-function conjectures. For all $s\ge1$ and $t\ge1$,

$$EN_{2,t}(321)[q]=q^t c_t(q),$$ $$EN_{s,2}(123)[q]=q^{2\binom{s}{2}}c_s(q),$$ $$NE_{s,2}(123)[q]=q^{s^2}c_s(q),$$ $$NE_{2,t}(123)[q]=q^{t^2}c_t(q).$$

These four identities are presented as conjectures and were verified computationally for $s\le9$ and $t\le9$; their general validity remains open.