I always thought that the Ramsey or Wilshere problem would be solved only when one of them leaves. Sort of how the Ox/Bellerin choice was resolved,
We shouldn't be choosing between Ramsey and Wilshere in the first place. A top club would expect both of them to compete in the squad. Anyway if it is a choice between the two, in footballing terms it's hardly sensible to pick Wilshere over Ramsey. As of yet Wilshere is a player who produces moments of quality play but does not control games or change games with end product, while Ramsey does have the end product at least.
Anyway, they should both stay. They shouldn't be nailed on starters, but they should be in the squad competing with each other under a new manager.