  It is possible to reduce the discrepancy between the local measurement of the
cosmological parameter $H_0$ and the value derived from the $Planck$
measurements of the Cosmic Microwave Background (CMB) by considering
contamination of the CMB by emission from some medium around distant
extragalactic sources, such as extremely cold coarse-grain dust. Though being
distant, such a medium would still be in the foreground with respect to the
CMB, and, as any other foreground, it would alter the CMB power spectrum. This
could contribute to the dispersion of CMB temperature fluctuations. By
generating a few random samples of CMB with different dispersions, we have
checked that the increased dispersion leads to a smaller estimated value of
$H_0$, the rest of the cosmological model parameters remaining fixed. This
might explain the reduced value of the $Planck$-derived parameter $H_0$ with
respect to the local measurements. The signature of the distant foreground in
the CMB traced by SNe was previously reported by the authors of this paper --
we found a correlation between the SN redshifts, $z_{\rm SN}$, and CMB
temperature fluctuations at the SNe locations, $T_{\rm SN}$. Here we have used
the slopes of the regression lines $T_{\rm SN}\,/\,z_{\rm SN}$ corresponding to
different {\it Planck} wave bands in order to estimate the possible temperature
of the distant extragalactic medium, which turns out to be very low, about
5\,K. The most likely ingredient of this medium is coarse-grain ($grey$) dust,
which is known to be almost undetectable, except for the effect of dimming
remote extragalactic sources.
