Converse characterization of the essential approximate point spectrum for elliptic and double parabolic Blaschke products

Let $B$ be an elliptic or double parabolic Blaschke product of degree $d \geq 2$, and let $T$ be the weighted composition operator on the disc algebra associated with $B$ and a weight $w$. Consider the set of moduli of scalars whose circles lie in the upper semi-Fredholm spectrum:

$$\{\lvert\lambda\rvert: \lambda \mathds{T} \subset \sigma_{usf}(T)\}.$$

Converse characterization conjecture. The set $\{\lvert\lambda\rvert: \lambda \mathds{T} \subset \sigma_{usf}(T)\}$ is at most countable and its only accumulation point, if any, is $0$.

The surrounding discussion presents this as the converse to the preceding theorem for finite Blaschke products. The supplied text does not establish the converse, so its resolution remains open.