On the non-uniqueness of q-cones of matroids

Let M be a rank-r simple GF(q)-representable matroid. A q-cone of M is a matroid M′ that is constructed by embedding M in a hyperplane of PG(r,q), adding a point p of PG(r,q) not on H, and then adding all the points of PG(r,q) that are on lines joining p to an element of M. If r(M)>2 and M is uniquely representable over GF(q), then M′ is unique up to isomorphism. This note settles a question made explicit by Kung by showing that if r(M)=2 or if M is not uniquely representable over GF(q), then M′ need not be unique.