The bones of the chest or rib-cage do sympathetically vibrate on lower pitches, and this is why we have the term “chest resonance” – a feeling of bone conduction of vibrating air molecules.
But, this does not enhance our sound to any appreciable degree above the lowest pitches. It would be erroneous when ascending the scale to maintain this lower vibrational feel. If one attempted to do so, then they would be keeping the cords thick and not letting them adjust for higher pitches.
Bottom line: the chest cavity does not make a good resonator due to its being a house for the lungs which are spongy. There is a better resonator – the larynx!