I'm not sure that's true. I think they're talking about how much they're in the Sun and I think I can construct a single counter example, r=2 h=1 for the cone,

I'm not sure that's true. I think they're talking about how much they're in the Sun and I think I can construct a single counter example, r=2 h=1 for the cone, r=1 for the sphere should be the same volume. For light at 90 degrees the area in the sun would be 2m^2 for the cone and pi m^2 for the sphere. That's not a perfect calculation so I don't know where the limits are here but I don't think they're talking strictly about surface area.