$\lambda$-invariant stability in families of modular Galois representations

Consider a family of modular forms of weight 2, all of whose residual $(\mathrm{m}\mathrm{o}\mathrm{d}p)$ Galois representations are isomorphic. It is well-known that their corresponding Iwasawa $\lambda$-invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these $\lambda$-invariants grow or remain stable.