Short paths in lemniscates: are two roots always joined by a path of length < 2 in $\{|f|<1\}$? (Erdős #1041)

Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[z]$ be a monic polynomial all of whose roots lie in the open unit disk, $\lvert z_i\rvert < 1$ for all $i$. Consider the lemniscate sublevel set $$E(f)=\{z\in\mathbb{C} : \lvert f(z)\rvert < 1\},$$ an open set that contains every root of $f$ (since $f(z_i)=0$). Must there always exist a rectifiable path of length less than $2$, lying entirely inside $E(f)$, that connects two distinct roots of $f$? Equivalently: either prove that for every such polynomial some pair of distinct roots is joined by an intra-lemniscate path of length $< 2$, or exhibit a polynomial (all roots in the unit disk) for which every path inside $\{\lvert f\rvert<1\}$ joining any two distinct roots has length at least $2$.

Acceptance. FULLY RESOLVES (counterexample side): an explicit polynomial $f$ with exactly specified roots or coefficients (algebraic or rational data, not floats), a certificate that all roots lie strictly inside the unit disk, and a rigorous, machine-checkable certificate (e.g., interval-arithmetic enclosure of the region $\{\lvert f\rvert<1\}$ with a verified geodesic-distance lower bound) that every path inside the region joining any two distinct roots has length $\ge 2$. Floating-point evidence alone does not suffice — the disproof requires certified numerics or exact analysis. FULLY RESOLVES (proof side): a complete proof that for every monic polynomial with all roots in the open unit disk, two distinct roots are joined by an intra-lemniscate path of length $< 2$; a machine-checked proof (Lean, e.g. building on the existing formal-conjectures statement) is preferred, else a full written proof with all steps. ADVANCES: (a) certified record configurations — polynomials with a verified lower bound on the minimal root-pair intra-lemniscate geodesic distance approaching $2$, with reproducible code; (b) a proof of the conjecture for a restricted class strictly beyond the cases cited in the background (e.g., a fixed degree above the claimed degree-$4$ case, all roots real, or roots with prescribed symmetry); (c) any proved uniform upper bound $C$ on the length of the shortest intra-lemniscate root-connecting path, with $C$ explicit. Deliver the polynomial + certification code, or the proof file.