Solved; the site's label is PROVED. Both questions are answered: Konyagin proved in 1994 that $m(f)\le n^{-1/2+o(1)}$ for all but $o(2^n)$ of the sign choices, the first bound below $1$ and so the first answer, yes, to the first question, after Kashin's bound $m(f)\le n^{1/2}(\log n)^{-1/3}$ of 1987; Konyagin and Schlag proved in 1999 that $m(f)$ is not typically smaller than a constant times $n^{-1/2}$, so the exponent $-1/2$ is optimal; and Cook and Nguyen proved in 2021 the exponential limit law of $m(f)$ at that scale (all refereed; the accepted claim pages are Kashin 1987, Konyagin 1994, Konyagin and Schlag 1999 and Cook and Nguyen 2021, the last of scope full). The derived standing is solved/answered: the first question is answered yes, and the second asks for the behavior of $m(f)$, which a limit law determines rather than proves or disproves. The site's commentary credits the first question to Kashin, which the account in Konyagin's paper contradicts, as the Current assessment records. The site's page links a formalized statement in formal-conjectures; no formal proof is built or audited in this repository.