A conjecture of Yu and Chen related to the Erdős–Lewin theorem

Yu and Chen (2022) conjectured that there exists a constant $c \gt 0$ such that every integer $n\ge 2$ can be represented as a sum of integers of the form $2^\alpha 3^\beta $, all of which are greater than $cn/\log n$ and none of which divides any other. This conjecture strengthens a theorem of Erdős and Lewin, and the lower bound in the above conjecture is optimal up to a constant. The purpose of this paper is to prove this conjecture.