Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder.

The map $j:n\mapsto n+1$ extends to an self-injection of $\beta\omega$, which itself restricts to a self-homeomorphism $\phi$ of $\beta^*\omega$.

In ZFC+CH, is it true that $\phi$ and $\phi^{-1}$ are not conjugate in $\mathrm{Homeo}(\beta^*\omega)$?

Indeed in Shelah's model ("forcing axiom"), in which CH fails, there exists a homomorphism $\mathrm{Homeo}(\beta^*\omega)\to\mathbf{Z}$ mapping $\phi$ to $1$. So the non-conjugacy of $\phi$ with $\phi^{-1}$ is consistent. But under CH, the group $\mathrm{Homeo}(\beta^*\omega)$ is simple (Rubin) so the non-conjugacy couldn't be attested by a homomorphism to $\mathbf{Z}$ as above.

*Note:* Boolean algebraic translation through Stone duality: consider the endomorphism of the Boolean algebra $2^\omega$ of subsets of $\omega$ given by $A\mapsto \{a\in\omega:a+1\in A\}$. It induces an automorphism $\Phi$ of the quotient Boolean algebra $2^\omega/\mathrm{fin}$, where $\mathrm{fin}$ is the ideal of finite subsets. Is (under ZFC+CH) $\Phi$ non-conjugate to its inverse in $\mathrm{Aut}_{\mathrm{Ring}}(2^\omega)$?

Indeed Stone duality yields (in ZFC) an isomorphism $\mathrm{Homeo}(\beta^*\omega)\to\mathrm{Aut}_{\mathrm{Ring}}(2^\omega)$ mapping $\phi$ to $\Phi$.

*Further comments:*

A side question is whether it is consistent with ZFC that $\phi$ and $\phi^{-1}$ are conjugate, I don't know either (but I'm primarily interested in the CH case).

Also in ZFC it is easy to check that $\phi$ is not conjugate to $\phi^n$ for any $n\ge 2$.

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