Sorry, but I doubt LH would sell BD to Virgin, nor do I see the logic in LH doing a deal with Virgin, either by acquiring SQ's stake (how long have they been trying to sell out??) or a joint-venture.
Even if LH wanted to sell BD entirely, I just don't think Virgin have the money to buy it. Also, LH's business model is based around feeding traffic into its long-haul networks at FRA and MUC etc. Why would LH want to divert traffic away from its own long-haul network into Virgin (which is also outside *A and the TATL JV with UA) unless it owned 100% of the company?
All I can see happening in the immediate future is LH using BD's slots where advantageous for more feeder traffic into its European hubs, and a bit of TLC to improve revenue management, the e-commerce operation etc. LX has done very well after its acquisition by LH, and hopefully BD will do so too. Personally, I think BD staff have a lot of reasons to be optimistic.