Answer: Your landscaper is right - leaving some "head space" will allow for deep watering (simply flood the boxes with water up to the top of the sides and allow it to trickle down to fully saturate the soil). If you don't like the look of 2-3" of wood showing, you can add an organic mulch (peat, spaghnum moss, etc.) all the way up to the top. The organic mulch will cover the bare soil but will still allow watering space and allow good water penetration.
Best wishes with your new planter boxes.
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