I don't think so yet - he's 'unofficially' her limiter - the other girls/Pandoras know that he wants to be Bridget's limiter (and thus off-limits) and that she's more or less dragging her feet to it because of her 'issues'
.
I think that's why Dana caused so much of a stir with Bridget since she wanted Aoi to become
her Limiter instead (impressed by the GAR, his ability and all).