Odd-prime transchromatic shearing conjecture for equivariant spectral sequences

Let $p$ be an odd prime, let $h=m(p-1)p^{n-1}$, and consider the $C_{p^n}$-action on the Lubin–Tate spectrum $E_h$ and the induced $C_{p^n}/C_p$-action on $E_{h/p}$. Let $t-s$ denote the usual filtration-adjusted degree in the relevant spectral sequences, and let $d_r$ denote the $r$-th differential.

Odd-prime transchromatic shearing conjecture. There is a shearing isomorphism

$$d_{p\cdot r-1}\leftrightsquigarrow d_r$$

between the $C_{p^n}$-spectral sequence of $E_h$ on or above the line of slope $p-1$ within $t-s\geq 0$, and the $C_{p^n}/C_p$-spectral sequence of $E_{h/p}$ within $t-s\geq 0$.

This would extend the cyclic transchromatic isomorphism phenomenon from the prime $2$ to odd primes. The source explicitly leaves some terminology vague, and no proof or resolution is supplied in the given context.