We evaluate a
Laurent expansion in dimensional regularization parameter $\epsilon=(4-d)/2$ of
all the master integrals for four-loop massless propagators up to transcendentality weight twelve, using a recently developed method
of one of the present coauthors (R.L.) and extending thereby results
by Baikov and Chetyrkin obtained at transcendentality weight seven.
We observe only multiple zeta values in our results.
Therefore, we conclude that all the four-loop massless propagator integrals, with any
integer powers of numerators and propagators, have only multiple zeta values
in their epsilon expansions up to transcendentality weight twelve.