OPEN

Let $A\subset \mathbb{N}$ be an additive basis of order $2$. Must there exist $B=\{b_1<b_2<\cdots\}\subseteq A$ which is also a basis such that
\[\lim_{k\to \infty}\frac{b_k}{k^2}\]
does not exist?

Erdős originally asked whether this was true with $A=B$, but this was disproved by Cassels [Ca57].