Mazur's conjecture on modular deformation liftings

Let $p$ be an odd prime, let $k$ be a finite field of characteristic $p$, and let deformation data be a pair $\mathcal D=(\Sigma,t)$ consisting of a finite set of primes and one of the conditions ordinary or flat. Let $\bar\rho:G_{\bold Q}\to\operatorname{GL}_2(k)$ be absolutely irreducible and $\mathcal D$-modular, meaning that it has a type-$\mathcal D$ lifting arising from an eigenform. A representation is type-$\mathcal D$ when it is a complete noetherian local $\bold Z_p$-algebra representation with cyclotomic determinant, unramified outside $\Sigma$, and satisfying $t$ at $p$. Mazur's conjecture. Every type-$\mathcal D$ lifting of $\bar\rho$ to the ring of integers of a finite extension of $\bold Q_p$ is modular. The source states that this conjecture implies the semistable modular lifting conjecture, and reports cases proved by Mazur and Ramakrishna for the ordinary and flat deformation conditions, respectively.