OED American English says it is. Plus there's no real historical precedent for it.

Apparently the whole 'shortsword' thing was invented by D&D nerds. Back in the day you had longswords and broadswords (which were often the exact same thing) and plain old swords. If it wasn't classified as 'long' or 'broad' then it was just called a sword, unless it had a fancy name of its own like 'rapier' or such.