  The measurements of $V_{us}$ in leptonic $(K_{\mu 2})$ and semileptonic
$(K_{l3})$ kaon decays exhibit a $3\sigma$ disagreement, which could originate
either from physics beyond the Standard Model or some large unidentified
Standard Model systematic effects. Clarifying this issue requires a careful
examination of all existing Standard Model inputs. Making use of a
newly-proposed computational framework and the most recent lattice QCD results,
we perform a comprehensive re-analysis of the electroweak radiative corrections
to the $K_{e3}$ decay rates that achieves an unprecedented level of precision
of $10^{-4}$, which improves the current best results by almost an order of
magnitude. No large systematic effects are found, which suggests that the
electroweak radiative corrections should be removed from the ``list of
culprits'' responsible for the $K_{\mu 2}$--$K_{l3}$ discrepancy.
