The bilinear identities conjecture for rectangular Schur superpolynomials

Let $k,n$ be integers with $k>1$ and $n>1$. Let $r',r\in\{k,k-1,0\}$ satisfy $r'>r$, and set $\epsilon=\delta_{r,0}$. Here $k^n$ denotes the ordinary partition with $n$ parts equal to $k$, and the semicolon separates the fermionic and bosonic parts of a superpartition. Rectangular bilinear identities conjecture.

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

These identities generalize the classical rectangular Schur bilinear identity and are proposed for the Schur superpolynomials; the source gives no resolution status.