Define a set of numbers with small radicals (A341645 in OEIS) by
$$A_2={ninmathbb{N} ;; text{rad}(n)^2le n}$$
The asymptotic density of $A_2cap {1,dots N}$ is $sqrt{N}times e^{2(1+o(1))sqrt{log N / log log N}}$, as per Lucia’s answer here.
 Main question: does the sumset $A=A_2+A_2$ contain all sufficiently large integers? In other words, is $mathbb{N}setminus A$ finite?
The number of misses is initially large but becomes sparse very rapidly. I didn’t find any after $86931723$, up to $10^9$. $A$ is not in OEIS (its complement is strictly a subset of A085253 there).
Other questions:

for any prime $p$, do the elements not divisible by $p$ have relative asymptotic density $0$ in $A_2$?

Computing (up to $10^9$) the subset $Bsubset A$ of sums of coprime pairs in $A_2$, points to the misses thinning out very slowly (still above $13%$ near $10^9$). Are there euristic arguments for or against $mathbb{N}setminus B$ being finite?
Is anything else known, or worth asking, about $A_2$, $A$ and $B$?