Disproved. The site's label is DISPROVED (LEAN), whose catalog suffix is explained under Formalization; the original texts are not held. The site's commentary credits the negative answer to de Caen and Székely [DeSz92]; their bounds are $n^{10/9}\gg f(n,\lfloor n^{2/3}\rfloor)\gg n^{58/57+o(1)}$ for $m\sim n^{2/3}$ and $f(n,m)\ll(nm)^{2/3}$ for $n^{1/2}\le m\le n$ (the general bound the site also attributes to Faudree and Simonovits), and the commentary records that Lazebnik, Ustimenko and Woldar [LUW94] later raised the lower bound to $f(n,\lfloor n^{2/3}\rfloor)\gg n^{16/15+o(1)}$. A $C_4$- and $C_6$-free bipartite graph with parts of sizes about $n^{2/3}$ and $n$ and $n^{1+\varepsilon}$ edges, $\varepsilon>0$ fixed, has more than $cn$ edges for every $c$ once $n$ is large, so no constant $c$ exists and the answer is no. Two claim pages record these results as accepted, on the site's acceptance and, for [LUW94], its refereed publication: de Caen and Székely and Lazebnik, Ustimenko and Woldar. Every exponent is second-hand: [DeSz92], a chapter of a 1992 Bolyai Society volume, and [LUW94], a journal paper whose Crossref record lists the publisher's open-archive license, are not held (the routes tried are recorded below). The external Lean file behind the site's suffix, which declares itself a formalization of de Caen and Székely's solution and is linked from their claim page (De Caen and Székely, 1992), proves, in its own statement, that for every $c>0$ some bipartite graph on $n$ vertices with a part of exactly $\lfloor n^{2/3}\rfloor$ vertices and at least $cn$ edges has no $6$-cycle, by the Lazebnik--Ustimenko--Woldar construction; it has not been built here. The standing derives from the two accepted claim pages, on the curator's credit for [DeSz92] and on the journal publication of [LUW94]; the Lean file is a formalization link, not evidence; and the exponents stay second-hand until [DeSz92] or [LUW94] is read at its theorem.