Set-wise independence conjecture for odd and even staircase Schur functions

For each $k\in{\mathbb N}$, let $\delta_k$ denote the staircase partition used in the paper, and let $s_{\delta_k}$ be its Schur function under the random specialization over ${\mathbb F}_q$. Odd-even staircase independence conjecture. The families

$$\{s_{\delta_k}\mid k\text{ odd}\}\quad\text{and}\quad\{s_{\delta_k}\mid k\text{ even}\}$$

are set-wise independent. Earlier results in the paper establish pairwise independence statements for separated staircases and for $s_{\delta_k}$ with $s_1=h_1=e_1$, but the displayed family-level assertion is not proved here.