  Hedetniemi's conjecture~\cite{hedetniemi1966homomorphisms} for $c$-colorings
states that the tensor product $G \times H$ is $c$-colorable if and only if $G$
or $H$ is $c$-colorable. El-Zahar and Sauer~\cite{El-ZaharS85} proved it for $c
= 3$. In a recent breakthrough, Shitov~\cite{Shitov19} showed counterexamples,
for large $c$. While Shitov's proof is already remarkably short, Zhu
\cite{Zhu20} simplified the argument and gave a more explicit counterexample
for $c=125$. Tardif \cite{Tardif20} showed that a modification of the arguments
allows to use ``wide colorings'' to obtain counterexamples for $c=14$, and
$c=13$ with a more involved use of lexicographic products. This note presents
two more small modifications, resulting in counterexamples for $c=5$ (with $G$
and $H$ having 4686 and 30 vertices, respectively).
