  We study estimation in the linear model $y=A\beta^\star+\epsilon$, in a
Bayesian setting where $\beta^\star$ has an entrywise i.i.d. prior and the
design $A$ is rotationally-invariant in law. In the large system limit as
dimension and sample size increase proportionally, a set of related conjectures
have been postulated for the asymptotic mutual information, Bayes-optimal mean
squared error, and TAP mean-field equations that characterize the Bayes
posterior mean of $\beta^\star$. In this work, we prove these conjectures for a
general class of signal priors and for arbitrary rotationally-invariant designs
$A$, under a "high-temperature" condition that restricts the range of
eigenvalues of $A^\top A$. Our proof uses a conditional second-moment method
argument, where we condition on the iterates of a version of the Vector AMP
algorithm for solving the TAP mean-field equations.
