Neck vibration can extract a little energy from the string, so a heavy or stiff neck might give you a little more sound than a lighter/'floppier' one. It's a little hard to say.
One thing that can effect the tone is the frequency of the 'neck' vibration, which is really a vibration of the whole guitar in which the neck bends a lot. If that 'neck mode' pitch matches the 'main air' resonance pitch the two will work together to move more air through the soundhole over a braoder frequency range. The tone of the lower notes can become particularly 'dark' and 'rich'.
To check the 'neck mode' frequency, hold the guitar up by pinching the neck right around the nut or first fret, so that the guitar hangs freely. Don't pinch too hard! Tap on the back surface of the headstock up at the upper ed, or else tap on the heel. Listen to the sound that comes off the headstock, and move your holding point around a little to get it to be as clear as possible. On many guitars it will be well below the pitch of the low E string, and thus won't effect the tone, but, if the neck is stiff enough, and the headstock and machines are not too heavy, it might be as high as F or G on the low E string.
The easy way to find the 'main air resonance' pitch is to lay the guitar down on it's back on a padded table top or in your lap. Pinch the low E string fairly tightly between your thumb and forefinger with the left hand, and pluck it with your right. Slide the pinch point up and down, listening for how loud the sound is. It will be muted, because of the damping of the flesh of your fingers. The loudest sound is the center of the 'main air' resonance, and will usually be near G on the low E string, although it can range a bit.
The frequency match between the 'neck' and 'air' modes has to be pretty exact to get the effect. Small changes in the stiffness of the neck, and the mass of the headstock, can change the 'neck mode' pitch and switch the effect on and off, which is why little things like swapping machines can make a big difference in the tone sometimes.
A carbon fiber rod will do a lot to stiffen up the neck. Tapering the neck, so that it's thicker by a few mm at the body end, also helps a lot.
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