Transpose-invariance and positivity conjectures for stable Schur expansions of degeneracy loci

Let $\Sigma^\wedge_{\infty,i}$ and $\Sigma^S_{\infty,i}$ denote the stable skew-symmetric and symmetric degeneracy loci, and write their Segre–Schwartz–MacPherson classes as formal Schur expansions $\sum_\lambda c_\lambda s_\lambda$, where $\lambda^T$ is the transpose partition. Transpose-invariance and positivity conjectures. The Schur expansions of $\operatorname{SSM}(\Sigma^\wedge_{\infty,i})$ and $\operatorname{SSM}(\Sigma^S_{\infty,i})$ are invariant under $\lambda\mapsto\lambda^T$: the coefficients of $s_\lambda$ and $s_{\lambda^T}$ are equal. Moreover, the Schur expansion of $\operatorname{SSM}(\Sigma^\wedge_{\infty,i})$ has non-negative coefficients, while the Schur expansion of $\operatorname{SSM}(\Sigma^\wedge_{\infty,2i})$ has alternating coefficients. These patterns arise from computed stable expansions and remain conjectural in general.