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Á. Elbert, Über eine Vermutung von Erdős betreffs Polynome I and II, Studia Sci. Math. Hungar. 1 (1966), 119--128, and 3 (1968), 299--324, prove the supremum half of the problem. For $f$ a nonconstant monic real polynomial with all roots in $[-1,1]$ and $E_f=\{x\in\mathbb R:|f(x)|<1\}$, the measure of $E_f$ is at most $2\sqrt2$; Erdős, Herzog and Piranian [EHP58] had proved that bound when every root is $\pm1$, where $(x^2-1)^m$ attains it, and conjectured it in general. The two papers are not held in the library, and the statement is recorded here as the later claimants report it: the manuscript of Darvas, Peng and Tao cites the two papers as the proof of the supremum, in a series of technical papers, and records that Erdős asked in 1976 for a more elementary proof, which Terence Tao's note of December 2025 supplies (Tao's page); Budala's Section 1.3 says that the sharp upper value was proved previously by Elbert. The site's commentary lists $\sup=2\sqrt2$ among the known values without naming the papers. No independent check of the proofs is recorded.

Covers. The supremum only: $\sup|E_f|=2\sqrt2$ over the class. The claim says nothing about the infimum, which the three pending full claims of the folder claim to determine.

Refereed. Studia Scientiarum Mathematicarum Hungarica, volume 1 (1966) and volume 3 (1968), as the manuscripts of Darvas, Peng and Tao and of Budala cite the two papers. No reviewer is named: the site's label is OPEN for the infimum, and its commentary credits no one with the supremum.

Depends on. Nothing on the wiki.