Conjecture on intermediate differentials in the genuine $C_2$-equivariant Adams spectral sequence

Let $M$ denote the algebraic Mahowald invariant for the 2-primary Adams spectral sequence. For integers $j$ and $k$ with $j\gg k>0$, let $s(k)$ and $c(k)$ be the stem and coweight of $M(h_0^k)$, respectively. The relevant values are

$$\begin{array}{c|c|c} k&s(k)&c(k)\\ \hline 4n+1&8n+1&4n\\ 4n+2&8n+2&4n\\ 4n+3&8n+3&4n\\ 4n+4&8n+7&4n+3 \end{array}$$

Intermediate-differential conjecture. For $j\gg k>0$,

$$d_{k+1}\left(\frac{\theta}{\rho^{2^{j-k}+s(k)}\tau^{2^j-1}}\right)=\frac{\theta}{\tau^{2^j+2^{j-k-1}+c(k)-1}}M(h_0^k)h_{j-k}.$$

This conjecture predicts a family of major intermediate differentials in the genuine $C_2$-equivariant Adams spectral sequence, extending the low-dimensional calculations discussed in the paper. Its status is unclear from the supplied text, so it remains open pending further verification.