The site labels the problem solved, with the two questions answered in opposite directions. The first question, whether a $K_4$-free graph of chromatic number $4$ has an odd cycle with at least two diagonals, is answered yes by the accepted partial claim Voss's two-chord theorem; the second, whether some $f(r)\to\infty$ diagonals are forced by local $3$-colorability, is answered no by the accepted partial claim the ten-chord construction of Alexeev, Putterman, Sawhney, Sellke and Valiant. The two claims together settle both parts of the problem, one yes and one no, so the problem is solved with the mixed value answered.