PS

*-šū̆ - his; him (genitive and accusative pronominal suffix, 3 masc. sg.) (Kogan 2009:69)

The shift *š- > h is distributed between languages the same way as in the realm of independent pronouns of the 3rd person.

Akkadian
-šu - his; him (genitive and accusative pronominal suffix, 3 masc. sg.) (GAG § 42 g)
Sargonic orthography distinguishes between genitive -SU and accusative SU₄.
Geez
-o, -hu - him (accusative pronominal suffix, 3 masc. sg.) (Bulakh–Kogan 2013b:101)
-hu is the allomorph used with verbal forms ending in *-ā (the etymological guttural is restored). See Bulakh–Kogan 2013b:105.
Geez
-u/-o, -hu - his (genitive pronominal suffix, 3 masc. sg.) (CDG 213, Bulakh–Kogan 2013b:101)
The form of the possessive suffix with nouns in nominative case is -u, while the allomorph used with nouns in accusative is -o. -hu is the allomorph used with forms ending in a vowel. See Bulakh–Kogan 2013b:103, 105.
Tigre
-u - his (genitive pronominal suffix, 3 masc. sg.) (Bulakh–Kogan 2013b:101)
Tigrinya
-u - his (genitive pronominal suffix, 3 masc. sg.) (Bulakh–Kogan 2013b:101)