I did that with a couple old boxes next to my house, it looked bad the first year-the deep cuts at any rate, the wood is pretty thick once you get into it. A year later, new growth has completely covered those spots. If it's healthy it should recover fairly well.Don't try too many undercuts-shade may slow or make new growth in these areas non-existant or irregular. A little fertilizer may help boost new growth as well. Give it a try on a section and see what happens, the boxes can be pretty forgiving.