Shortest escape path in $\{|f|\le 1\}$ from $0$ to the unit circle: worst-case growth in the degree (Erdős #1120)

Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$, and let $$E=\{ z : \lvert f(z)\rvert \leq 1\}.$$ (Note that $0\in E$, since $\lvert f(0)\rvert=\prod_i\lvert z_i\rvert\leq 1$.) What is the shortest length of a path in $E$ joining $z=0$ to the circle $\lvert z\rvert=1$? Writing $L(f)$ for this shortest length and $L(n)=\sup_f L(f)$ over all monic degree-$n$ polynomials with roots in the closed unit disc, the problem is to determine the worst-case behaviour of $L(n)$ as a function of $n$. Erdős wrote that 'presumably this tends to infinity with $n$, but not too fast'. His qualitative prediction that $L(n)\to\infty$ has recently been confirmed (see background); the surviving open problem is to pin down the order of growth of $L(n)$.

Acceptance. FULLY RESOLVES: matching upper and lower bounds determining the order of growth of $L(n)$ up to absolute constants, with complete proofs — for instance a bound $L(n)\asymp \phi(n)$ for an explicit $\phi$ lying between the currently known $\sqrt{\log n}$ and $n$. Machine-checkable (Lean) proof preferred, else a full written proof. ADVANCES: a lower bound strictly stronger than the known $L(n)\gg\sqrt{\log n}$ — e.g. an explicit family of degree-$n$ polynomials with reproducible certified $L(f_n)\geq c(n)$ growing faster than $\sqrt{\log n}$; an upper bound strictly better than the known $L(n)\leq \pi n$ (e.g. $L(n)=o(n)$) with proof; for concrete small $n$, explicit polynomials with a reproducible certified bound $L(f)\geq 1+\delta$ for quantified $\delta>0$, where the certification rigorously establishes (e.g. via interval-arithmetic enclosures of barrier regions where $\lvert f\rvert>1$) that every path in $E$ from $0$ to the unit circle has at least the claimed length; or exact/certified values of $L(n)$ for small $n$. Deliver the proof manuscript (or Lean file) and/or the constructions with certification code.