It appears that Wyoming also officially asked for these designations between Burgess Jct and Cody.
The Highway Commission minutes of 6/29/56 document a delegation from the Sheridan area requesting (among other things) that the US 14-WY 14 designations be replaced with US 14-N and US 14-S designations instead. The Commission said they had already unsuccessfully tried to get AASHO approval for that in the past (no date specified).
This is on pg. 63 of the 8th pdf at https://drive.google.com/drive/folders/1KPB9uJUfDzn0_G_1_Taba1OVgjuuvI7T
Then on pg. 73 of the same pdf the Commission agrees to try again with these designations to AASHO