Not exactly related but………..
Many years ago I had an appeal against a rent restriction and / or overpayment. We had paid £80.00 p/w (if (failing) memory serves me well) on a rent of £90 – £100 rent p/w. However, in evidence, the appellant supplied a rent book showing reduced charges whilst he had been working and not in receipt of HB. When questioned at interview he said that as he was working the landlady, who I beleive was a relative, accepted that he would struggle financially without the aid the HB payments so reduced the rent to £60.00 p/w. He had then lost his job and claimed again, and the rent had increased to £80.00 p/w which is the benefit payment he’d had before starting work.
I therefore made a determination (as it was then!) that the contractual rent was £60.00 p.w and always had been using the then Reg 7 clause of created to take advantage to apply only to the increase on the rent that occurred to relation to the benefit claim seeing that there was no way I would retrospectively prove the whole thing had been a sham (although I did suspect this). The decision was upheld by a review board and was not further challenged.
Applying that principle to your scenario I would consider the the first four weeks of the tenancy were created to take advantage of the scheme in that the LL would receive the rent on two properties and that from the end of the notice period you have a genuine liability. Therefore Reg 9 (1) (l) refers for the period of the overlap only as it was created to take advantage of Reg 7 (6) (d). This, of course, assumes you can get the evidence to back it up.
NB like the idea of not fit and proper though.
Do I know what I'm doing? The jury's out on that........................