Do the first $N$ cubes contain a Sidon set of size $\gg N$? (Erdős #1206)

A set is a Sidon set if all of its pairwise sums are distinct — equivalently, the only solutions to $a+b=c+d$ within it are the trivial ones. Does the set of cubes $\{1^3,2^3,\ldots,N^3\}$ contain a Sidon subset of size $\gg N$ (that is, of size at least $cN$ for some absolute constant $c>0$ and all $N$)? Equivalently, in the infinite formulation: is there an infinite set $A\subset\mathbb{N}$ of positive density such that $\{a^3:a\in A\}$ is a Sidon set?

Acceptance. FULLY RESOLVES (proof-shaped): a complete proof that $\{1^3,\ldots,N^3\}$ contains a Sidon subset of size $\geq cN$ for an absolute constant $c>0$ and all large $N$ (equivalently, that some positive-density $A\subset\mathbb{N}$ has $\{a^3:a\in A\}$ Sidon), or a proof that no linear-size Sidon subset / positive-density set exists. A machine-checkable proof is preferred, otherwise a complete written proof; a finite computation alone cannot establish the $\gg N$ (or infinite-density) claim. ADVANCES (each independently checkable): (a) improve the guaranteed Sidon-subset size for the cubes beyond the frontier stated in the background — prove $\{1^3,\ldots,N^3\}$ contains a Sidon subset of size $\geq N^{\theta}$ for some $\theta$ strictly exceeding the current record exponent (Gabdullin–Konyagin $1/2$, Garaev–Garayev–Konyagin $4/7-o(1)$), with proof; (b) compute the exact largest Sidon subset of $\{1^3,\ldots,N^3\}$ over a new range of $N$ with a reproducible search and an exhaustiveness certificate, extending the tabulated records. Deliver the proof, or the search code plus certified values.