Conjecture on mixed differentials in the \a0-motivic Bockstein spectral sequence

Let $x$ and $y$ be classes in $\operatorname{Ext}_{\mathbb{C}}$, with $x$ $\tau$-free and $y$ $\tau$-power torsion. Suppose that

$$d_r(x)=\rho^r y$$

in the ${\mathbb{R}}$-motivic $\rho$-Bockstein spectral sequence, and let $r<t<2^n$.

Mixed-differential conjecture. There exists a $\tau^{2^n}$-periodic differential

$$d_t(\tau^{2^n}x)=\rho^t z$$

if and only if the nonperiodic differential

$$d_{t-r}\left(\frac{Q}{\rho^{t-r}}y\right)=\frac{\gamma}{\tau^{2^n}}z$$

occurs in the $\rho$-Bockstein spectral sequence for the negative cone. If these differentials occur, there is a $\tau^{2^n}$-extension from $y$ to $\rho^{t-r}z$ in

$$\operatorname{Ext}_{\mathbb{R}}\left(\frac{{\mathbb{M}_2}^{\mathbb{R}}}{\rho^{t+1}}\right)$$

that is hidden by the $\rho$-Bockstein spectral sequence.

The conjecture relates mixed differentials in the ${\mathbb{R}}$-motivic Bockstein spectral sequence to differentials in the negative-cone Bockstein spectral sequence; the paper provides evidence for this relationship, but the general assertion remains open.