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Well Matthew has put his inheritance in from Lavinia's father, so I imagine he will take over
***SPOILER**** do not read further if you haven't watched it, but this is for Bednobs.
Lady Sybil went into labour there was a dispute between the local GP and the posh surgeon they called in. the local gp said she had eclampsia and needed emergency treatment, the posh surgeon over-ruled that as did Lord Grantham. She gave birth to a baby girl and then suffered convulsions and died. Lady G blames Lord G.