Entire functions with many maximum-modulus points: can $\liminf_{r\to\infty}\nu(r)=\infty$? (Erdős #1117)

Let $f$ be an entire function which is not a monomial, and for $r>0$ let $\nu(r)$ count the number of points $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\max_{\lvert w\rvert=r}\lvert f(w)\rvert$. (This is a finite quantity when $f$ is not a monomial.) Erdős asked two questions: is it possible that $$\limsup_{r\to\infty} \nu(r)=\infty\,?$$ Is it possible that $$\liminf_{r\to\infty} \nu(r)=\infty\,?$$ The first question was answered affirmatively by Herzog and Piranian (see background). The open problem is the second: does there exist an entire function, not a monomial, with $\nu(r)\to\infty$ as $r\to\infty$ — i.e. whose maximum modulus is attained at an unboundedly growing number of points on every sufficiently large circle?

Acceptance. FULLY RESOLVES: an explicit entire function $f$ (not a monomial) together with a complete proof that $\liminf_{r\to\infty}\nu(r)=\infty$ (machine-checkable Lean proof preferred, else a full written proof with all estimates); OR a proof that no such function exists, i.e. every entire non-monomial $f$ has $\nu(r)$ bounded along some sequence of radii $r\to\infty$. ADVANCES: a construction with quantitatively stronger properties than the approximate result of [GlPa24] stated in the background, with a precise statement of which property is strengthened and a proof; a resolution of the $\liminf$ question within a restricted class (e.g. entire functions of finite order, or gap series); or a proof that for every $k$ one can force $\nu(r)\geq k$ outside an exceptional set of radii of prescribed small (e.g. finite logarithmic) size. Numerical exploration alone does not qualify unless coupled to a rigorous statement. Deliver the construction and proof manuscript (or Lean file), with any supporting numerics as reproducible code.