Mod $p$ main conjecture for branches of Hida families

Let $\bar{\rho}:G_{\mathbf{Q}}\to\operatorname{GL}_{2}(k)$ be the residual representation, let $\mathcal H(\bar{\rho})$ be its Hida family, and let $Y$ be a branch of this family. Write $L_p(Y)$ for the mod $p$ $L$-function of $Y$, and let $\operatorname{Sel}_{\mathcal S(Y)}(\mathbf{Q}_{\infty},\bar{\rho})$ be the corresponding Selmer group over the cyclotomic extension. Mod $p$ main conjecture. The function $L_p(Y)$ is non-zero, the Selmer group $\operatorname{Sel}_{\mathcal S(Y)}(\mathbf{Q}_{\infty})$ is finite, and

$$\lambda\bigl(L_p(Y)\bigr)=\dim_k\operatorname{Sel}_{\mathcal S(Y)}(\mathbf{Q}_{\infty},\bar{\rho}).$$

Assuming the analytic and algebraic $\mu$-invariants of $\bar{\rho}$ vanish, the paper states that this is equivalent to the main conjecture for members of the Hida family.