Joyce's asymptotic exactness conjecture for the operator D

Let $M$ be a hyperkähler manifold that is AC of order $l\geq 1$, and let $k\geq -1$ be an integer. Let $x:M\rightarrow\mathbb{H}$ be a smooth function such that $D(x)$ is a $1$-form on $M$, where $D$ is the operator of Section 3.1. Suppose that

$$\nabla^aD(x)=O(t^{k-a-1})$$

for $a=0,1,2$, where $\nabla$ is the Levi-Civita connection on $M$. Asymptotic exactness conjecture. There exists a smooth function $y:M\rightarrow\mathbb{H}$ such that $D(x)=D(y)$ and $y=O(t^k)$. Moreover, if $x$ takes values in the imaginary quaternions $\mathbb{I}$, then $y$ can be chosen to take values in $\mathbb{I}$. This is an analytic conjecture modeled on results for harmonic functions of polynomial growth on asymptotically flat manifolds; the difficulty is to obtain the corresponding result for the operator $D$ on asymptotically conical manifolds.