I agree and would add the more the Shi’a are going to back the strong Shi’a horse in Iraq. So I would say the more the majority Shi’a Iraqi government engages and destroys the Shi’a militias, it will win support of Sunni and Shi’a Iraqis.
That, and the Grand Ayatollah Sistani (the most powerful Shi’a Cleric) condemned Sadr and his militia recently and underlined the Iraqi Government as the only LAW of the land (not militias).