Palindromicity and real-rootedness conjecture for two-by-n-by-(n+1) grids

For every positive integer $n$, let $[r]$ denote a chain with $r$ elements and let $\mathcal{N}_{[2]\times[n]\times[n+1]}(x)$ be the antichain generating polynomial of the product poset. Palindromicity and real-rootedness conjecture. The polynomial

$$\mathcal{N}_{[2]\times[n]\times[n+1]}(x)$$

is palindromic and real-rooted, and therefore $\gamma$-positive. This is supported by computed examples and is a specialization of the broader real-rootedness phenomenon discussed in the paper; it remains open.