I wouldn't. The crack goes past the E and A tuners. I would take those tuners off, gently open the crack with small steel wedges just enough to get glue all the way through the length of the crack. Then clamp it tight, wiping off the excess and let it dry for 24 hours. It will be stronger than the wood at that point.
Fold a pc of aluminum foil in half and work the glue in from both sides w/the syringe. You can pull sheets of foil through to spread the glue sometimes also - keep water handy b/c it's messy.