The surgery formulation of the degree-one map conjecture

Let $M=\mathcal{W}\cup_\mathcal{T}\mathcal{V}$, where $\mathcal{T}=\partial \mathcal{W}=\partial \mathcal{V}=\mathcal{W}\cap\mathcal{V}$ is a genus-$g$ surface. Let $\gamma_1,\dots,\gamma_g$ be non-separating simple closed curves in $\mathcal{T}$ such that $\mathcal{T}-\bigcup_{i=1}^g\gamma_i$ is connected and each $\gamma_i$ bounds a disjoint incompressible surface in $\mathcal{W}$. The surgery formulation of the degree-one map conjecture. If $N$ is the closed 3-manifold obtained by replacing $\mathcal{W}$ with a genus-$g$ handlebody $H$ such that every $\gamma_i$ bounds a disk in $H$, then

$$g(M)\ge g(N).$$

This reformulation expresses the degree-one map question as a question about surgery on a 3-manifold with connected boundary; it remains open together with the general degree-one map conjecture.