A priori bounds for asymptotic unimodular constraint manifolds

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a unimodular constraint manifold with negative chords that is asymptotic to a constraint manifold $\mathcal{T}$ tamed by $\Omega$. A priori-bounds conjecture. There are $C^{\infty}$ a priori bounds on every closed form $\omega\subset\mathcal{P}$ satisfying

$$[\omega]=[\Omega].$$

This extends the preceding almost-complex conjecture to general asymptotic constraint manifolds. The necessary taming condition is explained in the supplied context, but no resolution of this broader conjecture is given.