It is believed that Old Akkadian (at least) still retained the Semitic distinction of s₁, s₂ and s₃ and used different signs for syllables containing each of these. This is reflected by the transcription of those signs.
I know that Akkadian cuneiform eventually had four series of signs for different sibilants: S, Z, Ṣ, and Š. The third of these wasn't present in Sumerian (which only had three sibilant series) and was a fairly late development.
However, Old Akkadian (aka Sargonic) cuneiform was somewhat different than the later sorts, and represented a very different dialect than classical Babylonian or Assyrian. Did it ever distinguish more sibilants than these, such as a separate reflex of Proto-Semitic *s₂ (aka *ś aka *ł aka sin)? And if so, how? (As in, what signs did they use?)