Uniqueness conjecture for extended Bogomolny equations under weaker asymptotic conditions

Let $u$ be a solution of $(*)$, where $u_0$, $u_1$, $u_2$, and $u_3$ have the meanings defined in the surrounding construction. Suppose that, for some $\epsilon>0$,

• When $y\rightarrow +\infty$, $e^{-u}|P(z)|=O(y^{-\epsilon})$ and $|\nabla(u-y)|=O(y^{-\epsilon})$.
• When $y\rightarrow 0$, $u=u_0+O(y^\epsilon)$ and $|\nabla(u-u_0)|=O(y^{-1+\epsilon})$.

The uniqueness conjecture. Then $u=u_3$. The inequalities need not be uniform in $z$.