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Claim. Theorem 1.5 of Paul Pollack, Carl Pomerance and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, 379--398, states that the longest run of consecutive integers in $[1,x]$ on which $\phi$ is nonincreasing, and likewise the longest on which it is nondecreasing, has length

$$\frac{\log_3x}{\log_6x}+(\alpha-\gamma+o(1))\frac{\log_3x}{(\log_6x)^2},$$

with $\log_k$ the $k$-fold iterated logarithm, $\gamma$ Euler's constant and $e^\alpha=\prod_p(1-1/p)^{-1/p}$; Remark 8.1 of the paper notes that its lower-bound construction is strictly monotone. Hence $G(n)$, the largest $k$ for which some $m$ with $m+k\le n$ has $\phi(m+1)>\cdots>\phi(m+k)$, satisfies $G(n)\sim\log_3n/\log_6n$. The function $F(n)$ of Problem 415 requires the strictly decreasing pattern of length $F(n)$ to occur below $n$, so $F(n)\le G(n)=o(\log_3n)$, and $F(n)=(c+o(1))\log_3n$ fails for every constant $c>0$. Only the upper bound in Theorem 1.5 is needed for this; the paper does not itself discuss $F(n)$, and the deduction is the one the site's commentary draws. The source card pollack_et_al_2013_sets_monotonicity_euler_totient_function records the statement of Theorem 1.5 and Remark 8.1.

Covers. The first question, read with $c>0$ as the Formulation on the problem page states: $F(n)$ is not $(c+o(1))\log\log\log n$ for any positive constant $c$. Not covered: the second question, which pattern fails first, and the third, whether the natural ordering is the most likely, which the pending full claim on Chojecki's page addresses.

Acceptance. Refereed: The Ramanujan Journal, volume 30, issue 3 (2013), published online 19 September 2012. The site's commentary records that the asymptotic answers the first question in the negative, but the site labels the problem OPEN, so that commentary is not listed as reviewed evidence. The corpus has not reproved the theorem and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited paper.