Cohabitation: ESA consequences?

Saw a couple last week. Each has separate award of ESA with LCWRA (mental and physical health grounds - I think they’re possibly both vulnerable) and live in separate council flats in same block. Man also gets PIP both elements so I assume (they had no award letters with them) he will be getting SDP. They get HB and say they don’t pay CT. They want to know benefit consequences if they move in together (also mentioned plans to marry).
I’m anxious to ensure I give the right advice but am not confident. Can they make a couple claim for ESA because of the SDP award? Or does that apply only to the recipient? Is there any danger of UC here?
I’m not being lazy - I’ve tried to get my head round CPAG handbook on this but would really appreciate some help.

The important point is that claims for benefits such as irESA and HB are not made jointly, unlike UC and Tax Credits. The claim is made by a claimant who then either does or doesn’t have a partner.

The only significance of UC in this situation is that new claims for legacy benefits cannot be made - but no new claim is needed to resolve this situation. One of the couple would have to abandon their irESA award and become a partner on the other’s award - but this isn’t a new claim so can still be done even where SDP isn’t in play.

That said, on the face of it, they will be better off under UC when they move in together than they will be under legacy benefits. They could claim UC, having him carry over his LCWRA assessment, her claiming carers allowance and they would be about £1k per annum better off than under legacy with no conditionality at all - so they should probably think about claiming. Obviously, this isn’t a factor if there is any prospect of her getting a PIP award and bringing double-SDP into it.