For the 3 to merge would be impossible! The people with the power would certainly not allow a tie up between the 3! I think even BA-LH is pushing it! Add to that AA it would make a stupidly large carrier and cover over 1/2 the world. It really would be ridiculous.
Anyway, LH would have to leave Star Alliance or BA and IB Oneworld which is like asking Jeremy Clarkson to join forces with Cherie Blair.
Not a cat in hells chance