First, I know of no quantitative studies of the subject. There may be some that address the issue indirectly but I don't specifically know of any.

But empirical evidence does not need to be quantitative. There is plenty of evidence that the French lost Algeria when the French public lost faith in both Algerie Francaise and the French Army. There is also a lot of evidence that opposition to the Vietnam War aided the VC/NVA in their cause by influencing US policy and actual support to the RVN. The survey data do correlate with policy.

Finally, this issue is all part of the war for legitimacy which is fought in the country where the war takes place, the countries that support the "host government," and the "court of world opinion." (Sorry about the shorthand.)