Second moment of gaps among non-multiples of a sparse set: does the limit exist? (Erdős #489)

Let $A\subseteq\mathbb{N}$ satisfy $\lvert A\cap[1,x]\rvert=o(x^{1/2})$, and set $$B=\{n\geq 1:\ a\nmid n\ \text{for all }a\in A\},$$ the positive integers divisible by no element of $A$. Writing $B=\{b_1<b_2<\cdots\}$, is it true that $$\lim_{x\to\infty}\frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2$$ exists and is finite?

Acceptance. FULLY RESOLVES (proof-shaped): a complete proof — machine-checkable (Lean/Coq) preferred, else fully written — that for every $A$ with $\lvert A\cap[1,x]\rvert=o(x^{1/2})$ the limit $\lim_x\frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2$ exists and is finite; or a counterexample — an explicit sparse $A$ for which the limit fails to exist or is infinite, with proof. ADVANCES: a proof for a new class of sparse $A$ strictly generalizing the squarefree case $A=\{p^2\}$ proved by Erdős (for instance all $A$ supported on $k$-th powers, or all $A$ of counting function $O(x^{1/2-\delta})$), with proof; or a reproducible computation of the normalized second moment over an extended range for several concrete sparse $A$, with the program and certified numerics, giving evidence for or against the limit. Deliver the proof, or the code plus tabulated moments.