Optimal discrepancy $h(d)$ of a $\pm1$-coloring of $\mathbb{N}$ on APs of common difference $d$ (Erdős #177)

Find the smallest $h(d)$ such that the following holds: there exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$, $$\max_{P_d}\left\lvert \sum_{n\in P_d}f(n)\right\rvert\leq h(d),$$ where $P_d$ ranges over all finite arithmetic progressions with common difference $d$. In other words, determine the slowest-growing function $h(d)$ for which a single $\pm1$-coloring of the positive integers has discrepancy at most $h(d)$ on every AP of common difference $d$, simultaneously for all $d$.

Acceptance. FULLY RESOLVES: determine the correct order of growth of the optimal $h(d)$ — an explicit exponent $\alpha$ (or explicit function) with (a) a construction/proof that some $f:\mathbb{N}\to\{-1,1\}$ achieves $h(d)\ll d^{\alpha+o(1)}$ and (b) a proof that every $f$ suffers $h(d)\gg d^{\alpha-o(1)}$. Machine-checkable (Lean/Coq) proof preferred, else a complete written proof. ADVANCES: a proof strictly improving either side of the frontier stated in the background — an upper-bound construction with exponent strictly below Beck's, or a lower bound with exponent strictly above Roth's $1/2$; or a reproducible computational study of the finite truncation (for stated ranges of $N$ and $d\le D$, exact minimal achievable discrepancy values with SAT/ILP optimality certificates and code), clearly presented as evidence about the growth exponent rather than proof. Deliver the proof file, or the construction with proof, or the solver code + certificates + tables.