You can prove that? HS2 will make journeys more attractive, it creates journeys that would not have happened before. People will move further from London and drive to the station (and also price out locals needing housing)
I think it is a negative effect for the regions, not a positive one
Current population estimates for 2017 has the UK population at 66 million, by 2035 that's due to grow to 73 million (+10%).
We each currently travel 7,134 miles a year by car. That's 470bn miles by car as a population. If we all do the same miles (although it is true that our number of miles by car has been falling) that increases it to 520bn miles.
Given that most people do a lot of their miles at peak times, without provision for them to drive (i.e. extra roads) they would be looking for alternitve ways of getting around. If there's limited scope (as the trains get busier and busier) for them to opt for rail and more bus routes being cut due to subsidys being withdrawn along with the limitations of how far people car cycle what other options are there? Pretty much only flights.
Can I prove that this will definitely be the case, no, but then neither can you prove that it won't. However given the above I would suggest that it is a fairly likely outcome if we don't provide extra rail capacity.
However not all rail capacity is created equal. In that a Beeching reopening wouldn't help with the long distance rail capacity. If anything it would make it worse as more people would be able to get to the long distance services.
As I've said before, about 1 hour from London is fairly typical for ~10% of workers to work in London but for a lot of the other workers who work in that area work for companies which are based there for semi regular travel.
All those workers (whether they work locally or on London) then spend quite a lot of their money locally, which then benefits the local economy. Especially if they have families (children having swimming lessons, learning musical instruments, going to local attractions, buying clothes, etc.).
Sure HS2 will result in some people working in London and inflating house prices, but probably not by very much given how much a season ticket would cost.