The wheel-free Burling-control conjecture

A wheel is a graph consisting of an induced cycle of length at least $4$ and one additional vertex adjacent to at least three vertices on the cycle. A graph class is Burling-controlled in the sense defined in the source. Wheel-free Burling-control conjecture. The class of graphs with no induced wheel is Burling-controlled. This is proposed after Burling graphs refuted wheel-free chi-boundedness; it remains open in the supplied source.