Uniqueness conjecture for half-plane differentials with fourth-order poles

Let $X$ be a Riemann surface equipped with poles whose orders are all $4$, and prescribe the residues and leading order terms at those poles. Theorem provides a half-plane differential with this prescribed data. Uniqueness conjecture. When all the orders of the poles are $4$, the half-plane differential with prescribed residues and leading order terms is unique. This conjecture concerns the remaining uniqueness question for the construction of half-plane differentials: non-uniqueness is known for single-poled half-plane differentials on $\hat{\mathbb{C}}$ with poles of order greater than $4$, even when the residue and leading order coefficient agree.