  We study the behaviour of the minimal slope of Euclidean lattices under
tensor product. A general conjecture predicts that $\mu_{min}(L \otimes M) =
\mu_{min}(L)\mu_{min}(M)$ for all Euclidean lattices $L$ and $M$. We prove that
this is the case under the additional assumptions that $L$ and $M$ are acted on
multiplicity-free by their automorphism group, such that one of them has at
most $2$ irreducible components.
