Blocking sets meeting every line at most $C$ times: uniform over all projective planes? (Erdős #1159)

Determine whether there exists a constant $C>1$ such that the following holds: for every finite projective plane $P$, there is a set of points $S$ such that $$1\leq \lvert S\cap \ell\rvert \leq C$$ for every line $\ell$ of $P$. (A set of points meeting every line is called a blocking set; the question asks whether every finite projective plane — Desarguesian or not, of any order — admits a blocking set that meets no line more than a bounded number of times, with the bound uniform over all planes.)

Acceptance. FULLY RESOLVES: a proof that some absolute constant $C$ works for every finite projective plane — the argument must apply to all planes (not only Desarguesian ones) and all sufficiently large orders together with the finitely many small ones — as a complete written proof or machine-checkable; OR a proof of unboundedness: an explicit infinite family of planes together with a proof that every blocking set in the $n$-th member meets some line in at least $f(n)\to\infty$ points. A finite computation alone cannot close this. ADVANCES: an improvement of the Erdős–Silverman–Stein $O(\log n)$ bound stated in the background to a strictly slower-growing function, even restricted to a natural class such as Desarguesian planes $PG(2,q)$, with proof; a proof that a bounded $C$ suffices for $PG(2,q)$ for all $q$; exact values of the minimal achievable $\max_\ell \lvert S\cap\ell\rvert$ for every known plane of order up to a stated bound (at least through order 9), each certified by a witness set $S$ (checkable by a program against the plane's incidence data) plus an optimality certificate (SAT/ILP); or a resolution of the stronger pairwise-balanced-design version in either direction. Deliver the proof file, or the incidence data + search code + per-plane witness and optimality certificates.