Order of the longest similarly-ordered run of Farey fractions: is $f(n)\sim cn$? (Erdős #1005)

Let $\frac{a_1}{b_1}<\frac{a_2}{b_2}<\cdots$ be the Farey fractions of order $n\geq 4$ (all reduced fractions in $[0,1]$ with denominator at most $n$, listed in increasing order). Call indices $k<l$ similarly ordered if $(a_k-a_l)(b_k-b_l)\geq 0$, i.e. their numerators and denominators change in the same direction. Let $f(n)$ be the largest integer such that whenever $1\leq k<l\leq k+f(n)$ the fractions $\frac{a_k}{b_k}$ and $\frac{a_l}{b_l}$ are similarly ordered. Estimate $f(n)$; in particular, is there a constant $c>0$ with $f(n)=(c+o(1))n$ for all large $n$?

Acceptance. FULLY RESOLVES (OPEN, proof-shaped): a complete proof that the limit $c=\lim_{n\to\infty} f(n)/n$ exists and a determination of its value (settling van Doorn's $c=1/4$ conjecture either way). ADVANCES: rigorously improve either side of the current bracket — a lower bound $\liminf f(n)/n\geq c_1$ with $c_1>\tfrac{1}{12}$, or an upper bound $\limsup f(n)/n\leq c_2$ with $c_2<\tfrac{1}{4}$, each with proof; OR extend the exact computation of $f(n)$ (OEIS A386893) to a new record $n$ with reproducible code, sharpening the empirical estimate of $c$. Deliver the proof, the improved constant with its derivation, or the computation code together with the extended table.