Hormone-by-change sorting:
Pubic and axillary hair growth (pubarche): ANDROGEN-driven (adrenal DHEA and testosterone), in both sexes. NOT estrogen.
Menstruation: estrogen builds the proliferative endometrium that later sheds; estrogen-dependent.
Vaginal cornification: estrogen matures and cornifies the vaginal squamous epithelium (basis of the maturation index); estrogen-dependent.
Cervical mucus: estrogen produces the abundant, thin, ferning mid-cycle mucus; estrogen-dependent.
So three options are estrogen effects and only hair growth is androgen-mediated, making hair growth the correct 'NOT estrogen-dependent' answer.
Important note on the key: the crowd-recalled key marks menstruation, but that is medically incorrect because menstruation clearly depends on estrogen-primed endometrium, whereas pubarche is androgenic. The defensible answer is option a, hair growth.