Erdős Problem #1206 — Linear-sized Sidon subsets of cubes

Is there a constant $c>0$ such that, for every $N$, the cubes $1^3,2^3,…,N^3$ contain a Sidon subset of size at least $cN$? Equivalently in the infinite direction, is there a positive-density set $A⊆N$ for which ${a^3:a∈A}$ is Sidon?