  The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled
with $1,2,3$ so that no two adjacent vertices are incident to the same sum of
labels. In the last decades, several aspects of this problem have been studied
in literature, including more general versions and slight variations. Notable
such variations include the List 1-2-3 Conjecture variant, in which edges must
be assigned labels from dedicated lists of three labels, and the Multiplicative
1-2-3 Conjecture variant, in which labels~$1,2,3$ must be assigned to the edges
so that adjacent vertices are incident to different products of labels. Several
results obtained towards these two variants led to observe some behaviours that
are distant from those of the original conjecture.
  In this work, we consider the list version of the Multiplicative 1-2-3
Conjecture, proposing the first study dedicated to this very problem. In
particular, given any graph $G$, we wonder about the minimum~$k$ such that $G$
can be labelled as desired when its edges must be assigned labels from
dedicated lists of size~$k$. Exploiting a relationship between our problem and
the List 1-2-3 Conjecture, we provide upper bounds on~$k$ when $G$ belongs to
particular classes of graphs. We further improve some of these bounds through
dedicated arguments.
