Was the electroweak phase transition first order?
In plain words
In the SM, the Higgs field switched on smoothly as the early universe cooled. New particles could instead have made it an abrupt boiling transition, which could produce the matter excess and gravitational waves.
Precise statement
Lattice results show the SM electroweak transition is a crossover for $m_h = 125\,\mathrm{GeV}$. Determine whether nature has BSM scalars or couplings that make it strongly first order, $v(T_c)/T_c > \sim 1$. An answer identifies the order of the transition from measured Higgs potential parameters or excludes all first-order scenarios with a singlet or doublet extension.
What would settle it
A measurement of $\kappa_{\lambda}$ and of Higgs-singlet mixing at future colliders, or detection of a stochastic gravitational-wave background peaked near mHz by LISA.