Not exactly, but close.
The Shi'ites of Iraq are divided into two groups. By far the largest are Arab nationalist Shi'ites. They have their own grand ayatollah (Sistani). They are Arabs and Iraqis, and do not want their country to become a religious province of Iran's mullocracy. Shi'ism is not like Catholicism, with an absolute center in the Pope at Rome. It is more like Eastern Orthodoxy, with several patriarchs, each independent, with the Ecumenical Patriarch theoretically first among equals, but with the Greek Church and Russian Church not giving a fig what the leadership of the other Orthodox branch orders. They respect their religious brethren, but they do not OBEY them. That is like most Iraqi Shi'ites. They're ARABS, not Persians. Many are veterans of the war with Iran. They have a deadly grudge against the Sunni Arabs who murdered them (who are not Wahabbis, but the way), not the Sunni Kurds who didn't (and who are also not Wahabbis, but are of a different school of Islam than the Sunni Iraqi Arabs).
The biggest portion of Iraqi shi'ites are Shi'ite Iraqi Arab nationalists.
The smaller portion, but bnetter armed and organized, are the Shi'ite Iraqis who take their religious leadership from Iran. Al Sadr needs to be another tragic case of terminal kinetic energy poisoning.