Extension of Rudolph's mod-2 relation to decorated Kauffman satellite invariants

Let $L$ be a framed unoriented link with components $L_i$, and let $y_{\lambda_i}$ and $Q_{\lambda_i,\lambda_i}$ denote the corresponding Kauffman and Homfly annular decorations for partitions $\lambda_i$. The parameter satisfies $z=s-s^{-1}$ unless the relevant partition is self-conjugate. Extension of Rudolph's relation. Decorate each component $L_i$ of $L$ by $y_{\lambda_i}$. The Kauffman polynomial of this decorated link, with $v,s$ replaced by $v^2,s^2$ and the coefficients reduced modulo $2$, should equal the modulo-$2$ reduction of the Homfly polynomial of $L$ when each $L_i$ is decorated by $Q_{\lambda_i,\lambda_i}$. This would extend Rudolph's known case $|\lambda|=1$ to general Kauffman satellite invariants; the available evidence is limited, so the assertion remains open.