Cover the lemniscate $\{|f(z)|\le 1\}$ of any monic polynomial by discs of total radius $\le 2$ (Erdős #509)

Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the lemniscate set $$E_f=\{z\in\mathbb{C} : \lvert f(z)\rvert\leq 1\}$$ always be covered by a collection of discs the sum of whose radii is at most $2$? The constant $2$ would be sharp: on a real interval of length $\ell$ the least sup-norm of a monic degree-$d$ polynomial is $2(\ell/4)^d$ (attained by the rescaled Chebyshev polynomial), so taking $\ell=4\cdot 2^{-1/d}$ makes this sup-norm exactly $1$ and gives a monic $f$ whose set $E_f$ contains a real segment of length $4\cdot 2^{-1/d}\to 4$; since a disc of radius $r$ covers a segment of length at most $2r$, covering $E_f$ then forces total radius $\to 2$.

Acceptance. FULLY RESOLVES: a proof that for every monic non-constant $f\in\mathbb{C}[z]$ the set $E_f$ admits a covering by discs of total radius $\le 2$ (machine-checkable preferred, matching the formal-conjectures Lean statement, else a complete written proof); OR a counterexample: an explicit polynomial $f$ together with a rigorous proof (analytic, or computer-assisted with certified numerics) that every covering of $E_f$ by discs has total radius $>2$. ADVANCES: (a) an improvement of the general constant strictly below the $2.59$ bound stated in the background, with proof; (b) a proof of the constant $2$ for a natural class beyond connected lemniscates (e.g. lemniscates with at most $k$ components for some fixed $k\ge 2$, or all polynomials of degree $\le d$ for some explicit $d\ge 2$); (c) progress on Erdős's higher-dimensional generalisation (Problem 4.23 in [Ha74]) with an explicit constant; (d) a reproducible computational pipeline that, for input $f$, produces a certified disc covering of $E_f$ with rigorously verified total radius, applied to a documented family probing the disconnected regime. Deliver the proof file, or the polynomial plus certified lower-bound argument, or the covering code with certificates.