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Liverpool - Nottingham's don't serve Derby or Leicester, which were the places I cited.
Re-opening Matlock - Buxton would provide a link from Leicester, Loughborough and Derby to Manchester. It would remove traffic from Sheffield and make precisely no difference to Nottingham, which already has direct services to the NW.
And I restate, when Rio was in place, MML found that any incremental passengers to Manchester were more than off-set by those who wanted to travel from London to Sheffield or Nottingham.
If it was that attractive to have an E-Mids - Manchester service avoiding Sheffield, I think MML / EMT would have already pitched for it. As it stands they've focused on improving the Sheffield and Nottingham services......
You might have a fixation on Project Rio, however that doesn't negate the fact that reopening the Peak Forest route would have a transformative effect on travel between the North West and the whole of the East Midlands and beyond.
And Derby, Leicester and Loughborough are fairly sizeable settlements. Given decent marketing and enough time for people to get to know that it exists, I've little doubt that this could grow into a decent flow in itself. Maybe not enough to fill a HST, but a decent flow nevertheless.
I agree wholeheartedly with your point that lots of people want to go to Sheffield. This proposal would provide more capacity to go to Sheffield for those people, not less.
A last thought on Project Rio. It was an excellent idea, but from what I understand, it wasn't as heavily loaded from the Manchester area as was expected. This might have been a lack of publicity, perhaps Virgin didn't want people getting used to a rival TOC. Either way, if it didn't entirely reach the target audience it was marketed at, it's hardly likely to reach an entirely new passenger flow with much less promotion.