  Precision electroweak tests are a powerful probe of physics beyond the
Standard Model, but the sensitivity is limited by the precision with which the
$W$ boson mass ($M_W$) has been measured. The Parton Distribution Function
(PDF) uncertainties are a potential limitation for measurements of $M_W$ with
LHC data. It has recently been pointed out that, thanks to LHCb's unique
forward rapidity acceptance, a new measurement of $M_W$ by LHCb can improve
this situation. Here we report on a detailed study on the mechanism driving the
PDF uncertainty in the LHCb measurement of $M_W$, and propose an approach which
should reduce this uncertainty by roughly a factor of two using LHCb Run 2
data.
