Optimality conjecture for the Mayer-Erdős phenomenon on similarly ordered Farey fractions

Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey sequence of order $n$, and let $f(n)$ denote the least positive integer $r$ such that some pair of fractions $r$ positions apart is not similarly ordered. For $d=1,2,2,4$ according as $n\equiv 0,1,2,3\pmod{4}$, the upper bound is expected to be sharp. Optimality conjecture. For all $n\geq 4$,

$$f(n)>\frac{n}{4}.$$

More precisely, for all $n\geq 92$,

$$f(n)=\left\lfloor\frac{n}{4}\right\rfloor+d.$$

Here $d$ is defined by the residue class of $n$ modulo $4$ as above. The claim concerns the eventual optimality of the proven upper bound for the first non-similarly ordered pair of Farey fractions; the stated threshold and equality are supported by computer calculations in the source, but no proof or resolution is supplied.