Westminster rejected the referral because it thought Southwark had been wrong on the intentional homelessness issue, leaving Bantamagbari in a housing "no man's land".
On his claim for judicial review, the judge decided Westminster had acted unlawfully. It was bound by Southwark's decision that a housing duty was owed unless it could itself get that decision quashed in a judicial review. Although it had joined Southwark as a defendant to Bantamagbari's claim, Westminster had failed to demonstrate that the decision of unintentional homelessness was wrong.
Source
Housing Today
Reference
There is a detailed arbitration arrangement for disputes about which authority a homeless applicant has a connection with. It does not stretch to disputes about whether a duty is owed in the first place. Perhaps this case shows that it should.