Comparability sets in $[N]^3$ (Green Problem 88 / Gowers-Long)

For $x,y\in[N]^3=\{1,\dots,N\}^3$, call $x,y$ comparable if the difference $x-y$ has either two strictly positive coordinates or two strictly negative coordinates. Let $S\subseteq[N]^3$ have the property that every two distinct elements are comparable. Is $|S|\le N^{2-\delta}$ for some absolute $\delta>0$? Equivalently: bound the maximum independent set of the explicit graph on $[N]^3$ that joins $x,y$ whenever they are NOT comparable. A single 'layer' gives the trivial bound $|S|=O(N^2)$; the goal is to break $N^{2-\delta}$. Green also poses a symmetric modular version in $\mathbb{F}_p^3$: with six explicit 'comparability' classes forming a symmetric set $S$, if $A\subseteq\mathbb{F}_p^3$ has $A-A$ avoiding $S$, must $|A|\le p^{2-\delta}$?

Acceptance. ADVANCES: certified exact values of the maximum comparable-set size $\alpha(N)$ (with the extremal $S$ and an independent-set optimality certificate) for a range of small $N$, plus the fitted exponent and extremal structure; likewise for the $\mathbb{F}_p^3$ variant. ADVANCES: a rigorous sub-$N^2$ bound $|S|\le N^{2-\delta}$ (e.g. a weighted Delsarte-LP dual certificate breaking the spectral barrier), or a construction beating the naive single-layer lower bound. FULLY RESOLVES: a proof (or disproof) that $|S|\le N^{2-\delta}$ for some absolute $\delta>0$, correspondingly for the $\mathbb{F}_p^3$ version.