Verify the normalization of the prismatic logarithm comparison

Prove that the unit \(\alpha\in \mathbb{Z}_p\mathbin{\llbracket}q-1\mathbin{\rrbracket}^{\times}\) relating the Bott class to the prismatic logarithm satisfies \(\alpha=1\).

Background

The paper identifies the inverse Bott class $t^{-1}$ with a unit multiple of the prismatic logarithm $(q-1)^{-1}\log_{(q^p)}$, and consequently identifies the Bott class $\betaeta=(q-1)t^{-1}$ with a unit multiple of $\log_{(q^p)}$.

Equivariance shows that the multiplicative factor $\alpha$ is a unit in $\mathbb{Z}_p^{\times}$, but the paper does not determine this unit. The authors state that it should equal one, while also noting that this normalization is not needed for the results proved there.