  We show that the Markov semigroup obtained by Floricel in [Flo08] compressing
the $E_0$-semigroup of Skeide [Ske06], does not consist of endomorphisms. It,
therefore, cannot be the tail flow of an $E_0$-semigroup. As a corollary of our
result, Floricel's construction will allow to get examples of proper type III
Markov semigroups that are not tensor products of simpler ones, provided we
find type III Arveson systems that do not factor into tensor products.
