Symmetric anti-Ramsey number for odd cycles: settle the last open case $C_7$ (Erdős #809)

For a graph $G$, define the symmetric anti-Ramsey number $\chi_S(n,e,G)$ as the smallest $r$ for which there exists a graph on $n$ vertices with $e$ edges together with an $r$-colouring of its edges in which every copy of $G$ is *totally multicoloured* (all edges of the copy receive distinct colours). Taking the extremal edge count $e=\lfloor n^2/4\rfloor+1$, Burr, Erdős, Graham and Sós asked whether, for all $k\geq 3$, $$\chi_S\!\left(n,\ \lfloor n^2/4\rfloor+1,\ C_{2k+1}\right)\sim \frac{n^2}{8}.$$ Following recent progress this reduces to a single open case: does the asymptotic hold for the odd cycle $C_7$ (i.e. $k=3$)?

Acceptance. FULLY RESOLVES: settle the $k=3$ case — a proof that $\chi_S(n,\lfloor n^2/4\rfloor+1,C_7)\sim n^2/8$, i.e. matching bounds $(\tfrac18+o(1))n^2\leq \chi_S \leq (\tfrac18+o(1))n^2$; OR a disproof establishing $\lim_{n\to\infty}\chi_S(n,\lfloor n^2/4\rfloor+1,C_7)/n^2\neq \tfrac18$ (or that no such limit equals $\tfrac18$), with proof. ADVANCES: improve the bounds for the $C_7$ case beyond the general lower bound $\chi_S\gg_k n^2$ of [BEGS89] — e.g. prove an upper bound $\chi_S(n,\lfloor n^2/4\rfloor+1,C_7)\leq (c+o(1))n^2$ for an explicit constant $c$, or a lower bound $(c'+o(1))n^2$ with $c'$ improving on what is stated in the background, with proof; OR determine $\chi_S(n,\lfloor n^2/4\rfloor+1,C_7)$ exactly for a new range of small $n$ via exhaustive/SAT colouring search with a reproducible certificate (an explicit optimal host graph and colouring plus a certificate of optimality). Deliver the asymptotic proof, the improved bounds with proof, or the search code plus certified small-$n$ values.