May I recommend a trick from my guitar repairing days?
Get some araldite, the slow setting type, mix up a sufficient amount, then warm it with a hairdryer until all the air comes out (it will go transparent and runny). Then mix in some sawdust from the wood (sand a piece of scrap until you have enough. Mix it into the araldite, then warm it again. The advantage of using araldite over normal glue is that you can control the consistency by apply heat or allowing it to cool. When you apply the mix to the crack, heat it again and you will see bubbles come out as the air is expanded and driven out of the crack. You will need to add some more mix until there is a slight crown over the affected area, and applying heat no longer makes it shrink into the crack.
Leave it a few hours until it has started to set, then you can carefully slice off most of the excess with a sharp stanley blade. When full set, sand level.
For best results, experiment on a scrap piece first. Most sawdust will go a bit darker than the wood it came from, but that might be a good thing, certainly better than if it was a bit lighter
John