Can a minimal degree 6 cubature rule for the disk have all points inside?

We use positivity and extension properties of moment matrices to prove that a 10-node (minimal) cubature rule of degree 6 for planar measure on the closed unit disk D¯ cannot have all nodes in D¯. We construct examples showing that such rules may have as many as 9 points in D¯, and we provide similar examples for the triangle.