Conjecture on finiteness of Born transmission eigenvalues in bounded horizontal strips

Let $d=2$ or $d=3$, let $B_2$ be the ball of radius $2$, and let

$$m(x)=m(|x|)=\frac{(2+|x|)^{d-1}}{2^d|x|^{d-1}}.$$

A Born transmission-eigenvalue strip-finiteness conjecture. For any $C>0$, the strip

$$\{k\in\mathbb{C}:|\operatorname{Im}k|<C\}$$

contains at most finitely many Born transmission eigenvalues, where a Born transmission eigenvalue is a nonzero $k\in\mathbb{C}$ for which the associated Born transmission problem has a nontrivial solution. This is the formal version of the preceding strip-finiteness expectation; the theorem in the source proves discreteness and an eigenvalue-free strip near the real axis, but not finiteness in every bounded horizontal strip.