Erdős's additive irreducibility conjecture for perturbed squares

Let $R$ be obtained from the set of positive squares by deleting some of its elements up to $X$ and adding some positive integers, with the total number of changed elements being $o(X^{1/2})$. Erdős's conjecture. If $k\geq 2$ and we change $o(X^{1/2})$ elements of the set of squares up to $X$, then the new set $R$ is always additively irreducible. Here a set is additively irreducible when it cannot be written as a sumset $A+B$ with $A,B\subset\mathbb{N}$ both having size at least $2$. The conjecture remains open; Sárközy and Szemerédi proved it under the smaller perturbation bound $o(X^{1/2}/2^{(3+\epsilon)\log X/\log\log X})$ for every $\epsilon>0$.