Last summer I visited several old tree-girdling projects to see how well the technique worked, and I found that as many as 30% of the trees never died from the girdling. They recovered and kept on growing. But in most of those cases the issue was girdling technique; the person doing the girdling did only a single cut rather than two. That clearly isn't the case here. Even so, the tree could well live another year.
The bigger issue for this tree is that with a cut that deep and big, the tree could be more vulnerable to snapping off during high winds (the photo below is from a girdled tree that did just that). I'd steer clear of this tree on windy days.
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