Problem #881: Let $A\subset\mathbb{N}$ be an additive basis of order $k$...

Let $A\subset\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\subset A$ is any infinite set then $A\backslash B$ is not a basis of order $k$.

Must there exist an infinite $B\subset A$ such that $A\backslash B$ is a basis of order $k+1$?