Magnetic isoperimetric non-vanishing conjecture

Let $(M^m,g)$ be a compact manifold with smooth boundary $\partial M$, and let $\eta\in\Omega^1(M)$. Let $\mathfrak{B}_M$ denote the relevant gauge-trivial subspace, and for an admissible open subset $D\subset M$ let $\partial_I D$ be its interior boundary and $\iota^\eta(D)$ its magnetic isoperimetric quantity. Magnetic isoperimetric non-vanishing conjecture. If $\eta\notin\mathfrak{B}_M$, then there exist $\varepsilon,\delta>0$ such that every non-empty open subset $D\subset M$ with compact closure, smooth boundary, and

$$|D|\ge (1-\varepsilon)|M|,\qquad |\partial_I D|\le\varepsilon$$

satisfies

$$\iota^\eta(D)\ge\delta.$$

The conjecture would imply that the magnetic Cheeger constants remain strictly positive when the magnetic $1$-form is not in $\mathfrak{B}_M$, making the corresponding Cheeger inequality nontrivial. The supplied text gives no proof or resolution.