For an uncountable set of irrationals whose countable subsets sums all converge, is there one such countable subset which sums to a rational?

Question

This is a variation of Let $A\subset\mathbb{R}$ be an uncountable set of irrational numbers. Does there exist a finite $B\subset A$ such that $\sum_{x\in B} x\in\mathbb{Q}?$, relaxing the finiteness hypothesis on $B \subset A$ to countability, but demanding good behaviour on these with respect to summation