The tau-power conjecture for the rho-Bockstein spectral sequence

Consider the $\operatorname{\mathop{\rm Ext}}$ groups in the $\rho$-Bockstein spectral sequence

$$\bigoplus_{j\ge 0}\operatorname{Ext}_{\mathbb{C}}\left\{\rho^j\right\}\Rightarrow\operatorname{Ext}_{\mathcal{A}_{*,*}^{\mathbb{R}}}^{*,*,*}\left((H_{\mathbb{R}})_{*,*}\right).$$

Here, the power of $\tau$ is the exponent of $\tau$ in the relevant $\mathbb{F}_2[\tau^{\pm1}]\{\rho^j\}$ summand. Tau-power conjecture. Every differential in this $\rho$-Bockstein spectral sequence does not increase the power of $\tau$. Consequently, $[x]$ and $[\overline{y}]$ always have the same May's names. The conjecture would ensure that the lifting of differentials used in the comparison of the algebraic Atiyah–Hirzebruch spectral sequences is always valid; the authors state that they have neither a proof nor a counterexample.