Quillen's local parameter conjecture asserts that every finitely generated
projective module over $R[1/r]$ is free when $R$ is a regular local ring
and $r$ is a member of a regular system of parameters. Geometrically,
the problem concerns vector bundles on the principal open subset $D(r)$
of the local scheme $\Spec R$.
The assertion is known in all dimensions when $R$ contains a field,
by work of Bhatwadekar--Rao and Popescu; see
\cite[\S2.2.2]{CesnaviciusTorsorsSurvey}.
Gabber proved the case $\dim R\leq3$ in all characteristics, and Swan
gave an elementary proof~\cite{Swan1988}.
In mixed characteristic, known results in higher dimensions include the
case where $R$ is geometrically regular over a Dedekind subring containing
the chosen parameter~\cite[Theorem~1.3(2)]{CesnaviciusSmoothDivisor}.
This paper establishes the dimension-four assertion for arbitrary regular
local rings of mixed characteristic and arbitrary regular parameters,
without restrictions on ramification or residue characteristic.

The local parameter conjecture is related to the polynomial Bass--Quillen
problem by the dimension shift $Q_{n+1}\Rightarrow BQ_n$ in Swan's
notation~\cite[\S1]{Swan1988}. Here $Q_n$ denotes the local parameter
assertion in dimension $n$, and $BQ_n$ denotes freeness of projective
modules over a polynomial ring in one variable over a regular local
ring of dimension $n$.
The polynomial assertion is already known for regular local rings containing a field
and for unramified regular local rings of mixed characteristic; see
\cite[\S2.1.2]{CesnaviciusTorsorsSurvey}.
In dimension three, Rao proved it when the residue characteristic is
different from $2$ and $3$~\cite[Theorem~2]{Rao1988}.
Our result therefore implies the one-variable Bass--Quillen assertion
in dimension three, including the general ramified mixed-characteristic
cases with residue characteristic $2$ or $3$.