An ideal on $\omega_1$ is *strong* if it is precipitous and the associated generic elementary embedding always maps $\omega_1$ to $\omega_2$. This definition is from Baumgartner and Taylor, *Saturation Properties of Ideals in Generic Extensions II*
(available online at http://www.jstor.org/stable/1998900.) Every pre-saturated ideal on $\omega_1$ is strong (this was well-known even before the terminology was introduced, I think) and in this paper the authors ask whether the converse is true.

Does anyone know the status of this question: "is every strong ideal on $\omega_1$ pre-saturated"?