Cutting n-gons into triangles and quadrilaterals

Basically, we are trying to push the envelope beyond Monsky's theorem which states: a square cannot be cut into any odd number of equal area triangles.

Question: Given a convex n-gon. It is needed to cut it into equal area pieces that could be triangles or convex quadrilaterals but should not have more than 4 edges.

Claim: If for some integer m_0, a convex n-gon can be cut into m_0 equal area pieces that are all triangles or convex quads, then, it can be so partitioned into m pieces for all integer m > m_0.

The same question can be asked with perimeter replacing area.