Gδ-singleton criterion for hereditary realcompactness in ZF

Establish whether, in ZF, every realcompact space whose every singleton is a Gδ subset is hereditarily realcompact; equivalently, determine if the ZFC result that a realcompact space with all singletons Gδ is hereditarily realcompact remains provable without any form of the axiom of choice.

Background

Gillman–Jerison’s Corollary 8.15 (in ZFC) states that a realcompact space in which every singleton is Gδ is hereditarily realcompact. The authors do not know if this holds in ZF. They provide partial ZF results: if every singleton is a zero-set then hereditary realcompactness follows (ZF), and if every singleton is Gδ then hereditary realcompactness follows under CMC. The open question is whether the Gδ-singleton condition alone suffices in ZF.