Erdős Problem #33 — Minimal counting function of complements to the squares

Let $b_1<b_2<\ldots$ be an infinite sequence of integers such that every integer is of the form $b+k^2$ for some member $b$ of the sequence. Let $B(x)$ denote the number of members $b$ of the sequence with $b\leq x$. What is the smallest possible value of

$$\limsup_{x\to\infty}\frac{B(x)}{x^{1/2}}?$$

It is clear that $B(x)\geq x^{1/2}$ holds, but can one prove that

$$\liminf_{x\to\infty}\frac{B(x)}{x^{1/2}}>1?$$