HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω12 that is an HFD with the following property(P)the projection of H onto every partial product I2 for I∈ω[ω1] is onto. Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ2, there is an HFD topological group in ω12 which has property (P).