Imagine finding a pamphlet written by God that contains a listing of a perfect game of chess. Armed with it, you will still lose easily to Magnus Carlsen. He will deviate fro| the sequence and you won't know the perfect responses to his moves.
How would you go to prove that this is the perfect game of chess, if you didn't explore all of it's possible sequences? If they did solve the game, there's no way around knowing all possible outcomes - starting from a fixed given position.
An "ultra-weak solution" is even less. It gives the win/lose/draw outcome of the perfect strategy, without producing that strategy, nor even producing the game of perfect play from the initial state.
This is all covered in the second paragraph of the paper's introduction.
https://en.wikipedia.org/wiki/Solved_game
And if a strategy for perfect play only from the initial position is a strong solution, then what do you call the even more strongly solved case of a strategy for perfect play from any position?
The second paragraph of the paper isn't super specific about its definition for "weakly solved", but I read it as agreeing with my statement. It also calls checkers "weakly solved", a game for which a strategy to beat any possible move is known.
Even if you don't know the moves you will know that you are in a winning / drawn position. Magnus can not find a way to get to a winning position unless you make a mistake.
Perfect play in a perfect game means that the objective won't change - in optimal control theory this is called Pontryagin's Maximum Principle or the Bellman optimality criteria. A "value to go" function is enough to find the optimal solution.
That's the easiest part.