Modular degrees of elliptic curves

Modular degree is an interesting invariant of elliptic curves. It is computed by a variety of methods. After computer calculations, Watkins conjectured that given $E/\mathbb {Q} of rank $R, 2^R$ devides $\deg (\Phi)$, where $\Phi : X_0(N) \to E$ is the optimal map (up to isomorphism of E) and $\deg (\Phi)$ is the modular degree of E. In fact, he observed that $2^{R+K}$ should divide the modular degree with $2^K$ depending on W, where W is the group of Atkin-Lehner involutions, 
\mid W \mid = 2^{\omega(N)}, N$ is the conductor of the elliptic curve and $\omega (N)$ counts the number of distinct prime factors of N.  We have proved that $2^{R+K}$ divides $\deg (\Phi)$ would follow from an isomprphism of complete intersection rings of a universal deformation ring and a Hecke ring, where $2^K = \mid W^{\prime} \mid $, the cardinality of a certain subgroup of the group of Atkin-Lehner involutions.