If you haven't got a source, then it might as well just have been made up.
And by "no matter how many new houses are built" I think the issue is the discussed "ambitious" plans towards building an additional 20 thousand houses a year over existing construction of 180k P/A. That won't make a bit of difference when net imigration to the UK is 273k P/A, and that (obviously) excludes people born in the UK.
By any sane back of the envelope calculation we could fill several million additional houses right now.
Dusting off prefab plans and building 250k prefabs a year on top of existing construction would make a dent in the problem pretty much immediately.