Injectivity conjecture for Schrödinger-type elliptic growth

Let $D(t)$ be a family of domains described by elliptic growth processes with respect to operators $\Delta-u_1$ and $\Delta-u_2$. Equivalently, let $g_w$ and $g_w^*$ be Green functions of $\Delta-u_1$ and $\Delta-u_2$, respectively, with the same singularity, and suppose that

$$\partial_n g_w=\partial_n g_w^*$$

everywhere on $\partial D$.

Injectivity conjecture. If $D(t)$ is described by both processes, then $u_1=u_2$; equivalently, under the Green-function conditions above, $u_1=u_2$.

The conjecture asks whether the elliptic growth process determines the underlying operator. The source states that noninjectivity is known for radial permeability functions in Laplace--Beltrami growth, but no such examples are known for Schrödinger-type growth.