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The name of this trope is too generic for what it actually entails. Perhaps something straightforward like "Handkerchief Mask" or something more alliterative like "Handkerchief Headgear of Hiding" or otherwise something that specifically refers to the garment and its purpose would be more appropriate.
I think the Crayon Shin-Chan Gag Dub referred to it as a 'nose bra'. Considering it's not exactly clothes, that's a decent alternate title.
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How well does it match the trope?