Multiplicity-one conjecture for localized new homology

Let $\overline{\rho}:G_F\rightarrow\operatorname{GL}_2(k)$ be an absolutely irreducible Galois representation, let $\mathfrak{m}$ be the associated maximal ideal, and let $d_F$ be the discriminant of $F$. Assume that $p$ divides neither $2$ nor $d_F$, and that for every $\mathfrak{q}\in S$, either $\overline{\rho}|G_{\mathfrak{q}}$ is ramified at $\mathfrak{q}$, or it is unramified there and $\overline{\rho}(\operatorname{Frob}_{\mathfrak{q}})$ is not scalar. Multiplicity-one conjecture. Then

$$H_1(Y(K_{\Sigma}),\mathbf{Z})^{\mathrm{new}}_{\mathfrak{m}}$$

is free of rank one over

$$\mathbf{T}^{\mathrm{new}}_{\mathfrak{m}}.$$

This is a localized multiplicity-one statement; the source notes that failures of multiplicity one can arise from local phenomena, especially at primes above $2$.