  The SH0ES collaboration Hubble constant determination is in a ${\sim}5\sigma$
difference with the $Planck$ value, known as the Hubble tension. The accuracy
of the Hubble constant measured with extragalactic Cepheids depends on robust
stellar-crowding background estimation. Riess et al. (R20) compared the light
curves amplitudes of extragalactic and MW Cepheids to constrain an unaccounted
systematic blending bias, $\gamma=-0.029\pm0.037\,\rm{mag}$, which cannot
explain the required, $\gamma=0.24\pm0.05\,\rm{mag}$, to resolve the Hubble
tension. Further checks by Riess et al. demonstrate that a possible blending is
not likely related to the size of the crowding correction. We repeat the R20
analysis, with the following main differences: (1) we limit the extragalactic
and MW Cepheids comparison to periods $P\lesssim50\,\rm{d}$, since the number
of MW Cepheids with longer periods is minimal; (2) we use publicly available
data to recalibrate amplitude ratios of MW Cepheids in standard passbands; (3)
we remeasure the amplitudes of Cepheids in NGC 5584 and NGC 4258 in two HST
filters ($F555W$ and $F350LP$) to improve the empirical constraint on their
amplitude ratio $A^{555}/A^{350}$. We show that the filter transformations
introduce an ${\approx}0.04\,\rm{mag}$ uncertainty in determining $\gamma$, not
included by R20. While our final estimate, $\gamma=0.013\pm0.057\,\rm{mag}$, is
consistent with the value derived by R20 and is consistent with no bias, the
error is somewhat larger, and the best fitting value is shifted by
${\approx}0.04\,\rm{mag}$ and closer to zero. Future observations, especially
with JWST, would allow better calibration of $\gamma$.
