"That came down on the edge of his robes" (133:2). What came down on his robes: the oil, or Aaron's beard?

The verse can be read either way, in the WEB and in Hebrew. The WEB's line is "that ran down on the beard, even Aaron’s beard, that came down on the edge of his robes" (Psalm 133:2), and that came down follows Aaron’s beard but can also go back to the precious oil. In Hebrew came down is the same verb as ran down (tap came down), and its form fits either noun: Brown-Driver-Briggs lists oil (shemen) as masculine, and beard (zaqan) as masculine and feminine. Edge is peh, a word Brown-Driver-Briggs gives as "mouth", and among its senses "opening, orifice" (tap edge). Exodus uses it for the opening of "the robe of the ephod" (Exodus 28:31) for the head: "It shall have a hole for the head in the middle of it" (Exodus 28:32), and "The opening of the robe in the middle of it was like the opening of a coat of mail" (Exodus 39:23), where hole and opening are peh. But the psalm's word for robes is another word, middah (tap robes), and it doesn't say which edge it means. These notes set out the words, and don't choose whether the oil or the beard came down on the robes.