  We consider the electroweak phase transition in the conformal extension of
the standard model known as SU(2)cSM. Apart from the standard model particles,
this model contains an additional scalar and gauge field that are both charged
under the hidden SU(2)$_X$. This model generically exhibits a very strong phase
transition that proceeds after a large amount of supercooling. We estimate the
gravitational wave spectrum produced in this model and show that its amplitude
and frequency fall within the observational window of LISA. We also discuss
potential pitfalls and relevant points of improvement required to attain
reliable estimates of the gravitational wave production in this - as well as in
more general - class of models. In order to improve perturbativity during the
early stages of transition that ends with bubble nucleation, we solve a thermal
gap equation in the scalar sector inspired by the 2PI effective action
formalism.
