https://en.wikipedia.org/wiki/Current_members_of_the_United_States_House_of_Representatives

Currently, Democrats hold majorities in the House delegations from 14 states. Republicans, 33. The remaining 3 are evenly split.

So in order to win a contest like this, we would need to flip 12 states (and hope they don't flip any back!). Note that each state we might hope to flip would require at least one congressperson to to completely surrender their lifelong views and career prospects. So it would not be an easy thing to do.

But hey, let's be optimistic.

Imagine we can flip the three evenly-split states. That brings our number up to 17.

Among the Republican-majority states, I counted seven (AK, AZ, MT, NE, ND, SD, WY) where flipping a single member would change the majority).

That ludicrously optimistic scenario puts us at 24 states. Still two short of a majority.

If the Democrats are split, the Republicans win.