Permanent-cycle conjecture for $a_{\sigma_n}u_{\rho_n-\sigma_n}$

Let $C_{2^n}$ be the cyclic group of order $2^n$, and let $a_{\sigma_n}$ and $u_{\rho_n-\sigma_n}$ denote the Euler and orientation classes associated to the representations $\sigma_n$ and $\rho_n-\sigma_n$. Consider the slice spectral sequence for $C_{2^n}$. Permanent-cycle conjecture. The element

$$a_{\sigma_n}u_{\rho_n-\sigma_n}$$

is a permanent cycle in the slice spectral sequence for $C_{2^n}$. Computational evidence supports this for $n=3$, but the authors note that the conjecture may be too strong: for larger $n$, differentials could occur, possibly reflecting Adams differentials.