It seems to disagree with your statements.
"In conclusion, in this primary prevention trial, we observed that an energy-unrestricted Mediterranean diet, supplemented with extra-virgin olive oil or nuts, resulted in a substantial reduction in the risk of major cardiovascular events among high-risk persons. The results support the benefits of the Mediterranean diet for the primary prevention of cardiovascular disease."
Another random snippet from same link: "Thus, extra-virgin olive oil and nuts were probably responsible for most of the observed benefits of the Mediterranean diets."