Hey,
As described in the last section of the README.md: https://github.com/joaquimg/BilevelJuMP.jl#conic-lower-level
It can model second order cone problem, but they are way more challenging.
First thing you need is to force the BilevelJuMP mode to BilevelJuMP.ProductMode as in:
model=BilevelModel(Gurobi.Optimizer, mode = BilevelJuMP.ProductMode(1e-5))