Erdős Problem #880 — Bounded gaps between restricted sums from an asymptotic basis

Let $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$ be an asymptotic basis of order $k$, so every sufficiently large integer is a sum of at most $k$ elements of $A$. If $B=\{b_1<b_2<\cdots\}$ consists of the integers representable as sums of at most $k$ distinct elements of $A$, must $\limsup_{i\to\infty}(b_{i+1}-b_i)<\infty$?