The Republic of Gilead is named for the biblical name "Gilead" for a region in Jordan[0]; the shot is presumably named for the Balm of Gilead[1], also named for biblical Gilead.
Descovy and Truvada are already more commonly referred to by their marketing names than with Gilead’s. Hard to see how this new shot would be different.
The trade name of the drug is Sunlenca; the generic name is lenacapavir.