Green’s Dictionary of Slang

(US black) to straighten hair with a mixture based on Congolene, thus adj. konked-down.

1942		‘Digg Mee’ ‘Observation Post’ in N.Y. Age 29 Aug. 9/8: He’ll appear from out of nowhere [...] rainin’ [sic] cain with my konked-down hair [...] [K]onkin’ hair ain’t no fun [...] Don’t be so mean. Please stop meltin’ my konko-lene.
1944		Chicago Defender 21 Oct. 12/7: Others conk their hair down and it gleams as brilliantly as if they were born in India and it looks fine.
1944		N.Y. Age 19 Aug. 9: My grass is konked and slick [...] He’ll [i.e. the sun] appear from out nowhere, raising hell with my konked-down hair.
1968		N. Heard Howard Street 149: He looked upon hair-conking with scorn.
1975		E. Torres Carlito’s Way 12: You had a pimp name of Red Conk on account of he conked his hair red.
1996		Eble Sl. and Sociability 82: Many terms that originated among African Americans remain for the most part in that community, like conk ‘to straighten hair’.