  Neutrino are massless in the Standard Model. The most popular mechanism to
generate neutrino masses are the type I and type II seesaw, where right-handed
neutrinos and a scalar triplet are augmented to the Standard Model,
respectively. In this work, we discuss a model where a type I + II seesaw
mechanism naturally arises via spontaneous symmetry breaking of an enlarged
gauge group. Lepton flavor violation is a common feature in such setup and for
this reason, we compute the model contribution to the $\mu \rightarrow e\gamma$
and $\mu \rightarrow 3e$ decays. Moreover, we explore the connection between
the neutrino mass ordering and lepton flavor violation in perspective with the
LHC, HL-LHC and HE-LHC sensitivities to the doubly charged scalar stemming from
the Higgs triplet. Our results explicitly show the importance of searching for
signs of lepton flavor violation in collider and muon decays. The conclusion
about which probe yields stronger bounds depends strongly on the mass ordering
adopted, the absolute neutrino masses and which much decay one considers. In
the 1-5 TeV mass region of the doubly charged scalar, lepton flavor violation
experiments and colliders offer orthogonal and complementary probes. Thus if a
signal is observed in one of the two new physics searches, the other will be
able to assess whether it stems from a seesaw framework.
