OSM is not an official source so I wouldn't pay what they say mind. What officially exists on paper is discernable here:
https://vegasja.vegagerdin.is/eng/Officially, TH410 does not extend north to TH413. And the sign you noted which was present in 2013 was subsequently removed: no newer GMSV imagery but Mapilary shows a new sign with no route number there by 2016:
https://www.mapillary.com/app/?pKey=408150843863518So, leaving that as is.
For TH413, checking my notes, I cut it short because it isn't signed from TH40. Didn't notice the sign you linked to shortly east of there. But it was and still is shown on the official map as extending to TH40, and that sign was still there last year, so I'm in agreement with extending it.