The bilinear identities conjecture for almost rectangular Schur superpolynomials

Let $k,n$ be integers with $k>1$ and $n>1$, and use the semicolon to separate the fermionic and bosonic parts of a superpartition. Almost rectangular bilinear identities conjecture.

$$\begin{gathered} s_{(k,k-1;k^{n-2})}s_{(k^n)}=-s_{(k+1,k;(k+1)^{n-2})}s_{((k-1)^n)}+s_{(k-1;k^n)}s_{(k;k^{n-2})},\\ s_{(k-1,0;k^{n-1})}s_{(k^n)}=s_{(k-1,0;(k+1)^{n-1})}s_{((k-1)^n)}+s_{(k-1;k^n)}s_{(0;k^{n-1})},\\ s_{(k,k-1;k^{n-2})}s_{(0;k^n)}=s_{(k+1,k,0;(k+1)^{n-2})}s_{((k-1)^n)}+s_{(k-1,0;k^n)}s_{(k;k^{n-2})},\\ s_{(k,k-1,0;k^{n-2})}s_{(k^n)}=s_{(k+1,k;(k+1)^{n-2})}s_{(0;(k-1)^n)}+s_{(k-1,0;k^n)}s_{(k;k^{n-2})}. \end{gathered}$$

These further identities concern almost rectangular super-diagrams for which neither the underlying partition nor its circle-completed partition is rectangular; the source gives no resolution status.