Green’s Dictionary of Slang

conk n.2

[Congolene, a fiery liquid, combining lye, eggs, potatoes and other ingredients, used in the artificial straightening of naturally kinky black (racially) hair]
(US black)

1. pomade; hair grease.

[US]Pittsburgh Courier (PA) 9 Sept. 11/1: Brother Jones has cropped his hair short to save the [cash] he once spent on ‘Konk’ and now his moss isn’t glossy any more .
[US] ‘Jiver’s Bible’ in D. Burley Orig. Hbk of Harlem Jive.

2. hair that has been straightened through the application of a special mixture.

[US]C. Himes ‘Let Me at the Enemy’ in Coll. Stories (1990) 36: This George Brown was strictly an icky, drape-shaped in a fine brown zoot with a pancho conk slicker’n mine.
[US]H. Simmons Corner Boy 14: The little man with the conk tossed Jake a dollar.
[US]E. Torres Carlito’s Way 9: You had a pimp name of Red Conk on account of he conked his hair red.
[US](con. 1986) G. Pelecanos Sweet Forever 44: French cuffs . . . man had a conk and shit.
[US]‘Touré’ Portable Promised Land (ms.) 156: We Words (My Favorite Things) [...] Chump. Crunk. Conk.

In compounds

conkhead (n.)

1. (US) one who uses hair oil.

[US]letter in N.Y. Age 29 Dec. 9/4: Nothing naseates me more than the sight of these ‘conkheads.’ I often wonder how much barrel of grease they must ladle onto their skulls.

2. (US black) a black person.

Hal Ellson ‘Decoy’ in Tell Them Nothing (1956) 40: A conkhead who don’t know he’s living says [etc.]. [Ibid.] ‘I Didn’t See a Thing’ 141: Two men. Bug-Eye and a conkhead.
conk job (n.)

(US black) a black hairstyle that has been straightened.

[US]G. Tate ‘Atomic Dog’ in Flyboy in the Buttermilk (1992) 29: My man was chemically processing hair – mixing down wave, curl, and conk jobs for all the slick splib hustlers.