It is known $P=BPP$ is insufficient to derandomize $VV$ isolation lemma.
What does it mean to be 'insufficient' here? Is there some theorem which says $P=BPP$ $+$ 'condition $A$' gives derandomization of Valiant-Vazirani.
Of course $P=NP$ suffices as condition $A$. However I am looking for anything weaker that would be possible and if something weaker than $P=NP$ is not possible then why?