The exploded Dolbeault-bundle conjecture for stable smooth curves

Let $\mathfrak B$ be an exploded $\mathbb T$ fibration with a closed $2$-form $\omega$ and almost complex structure $J$. Let $\mathfrak M^{sm}_{g,n}(\mathfrak B,\omega)$ be the moduli space of $\omega$-stable smooth exploded curves, and let $\mathfrak E_{g,n}(\mathfrak B,\omega)$ be the space of antiholomorphic sections of $T^*\mathfrak C\otimes f^*(T\mathfrak B)$ that vanish on fibers over edges of $\mathfrak C$.

The exploded Dolbeault-bundle conjecture. The space $\mathfrak E_{g,n}(\mathfrak B,\omega)$ has the structure of a Fréchet orbifold exploded fibration and is a vector bundle over $\mathfrak M^{sm}_{g,n}(\mathfrak B,\omega)$. The $\bar{\partial}$ equation defines a smooth section of this vector bundle.

This is the bundle-theoretic part of the paper's proposed perturbation theory. The source gives no resolution evidence for this asserted structure.