Sticky Matroid Conjecture

Let $M$ be a matroid. It is sticky if, for any two matroids $N_1$ and $N_2$ with $E(M)=E(N_1)\cap E(N_2)$ and $M=N_1|_{E(M)}=N_2|_{E(M)}$, there exists an amalgam of $N_1$ and $N_2$. A matroid is modular when every pair of flats is modular. Sticky Matroid Conjecture. Every sticky matroid is modular. This is the converse of the known fact that every modular matroid is sticky. The paper relates this conjecture to the hypermodular matroid conjecture and Kantor's conjecture.