The site labels the problem DISPROVED (LEAN), crediting Rödl's construction of nearly bipartite graphs of large chromatic number [Ro82] as noted in the thread; the Lean behind the qualifier is described under Formalization. The accepted claim is Rödl's nearly bipartite graphs, which answers both questions in the negative with the subgraph condition read, as the statement writes it, as chromatic number $\leq r$.