Much less so as bridge mass increases in solidbody instruments. The body's tonal importance in extreme cases is all but gone. That much we know.The wood of an electric solidbody is absolutely a major part of the instrument's sound. Different types of wood and sizes of body give very different sounds.
And I defy anyone to say with any exactitude how much of a particular instrument's "tone" is due to which components. Can't be done.
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