Let $X$ be a compact $4$-manifold, possibly with boundary.

Theorem 17.1.2 of Kronheimer-Mrowka's book "Monopoles and Three-Manifolds" states

Let $X' \subset X$ be a codimension-zero submanifold with boundary, contained in the interior of the manifold $X$. The restriction map $L^2_k(X) \to L^2_k(X')$ is a surjective map with continuous left inverse.

I was wondering if such an extension statement for solutions to the Seiberg-Witten equations could be found anywhere in the literature.

Specifically, can we fix boundary conditions $P$, $P'$ on $X$, $X'$ such that:

There is a well-defined restriction map from $\mathcal{M}(X, P)$ to $\mathcal{M}(X', P')$, the spaces in question being the gauge-equivalence classes of solutions to the Seiberg-Witten equations satisfying the boundary conditions.

This restriction map is surjective and admits a continuous left inverse.