Commuting-limits conjecture for the regularized one-dimensional point interaction

Let $\beta$ be the expansion parameter, let $a$ be the regularization parameter, and let $\psi_{a,\beta}(x)$ denote the function given by Theorem. At fixed $a$ and $x$, consider an expansion in $\beta$. Commuting-limits conjecture. The function $\psi_{a,\beta}(x)$ admits an expansion

$$\psi_{a,\beta}(x)=\psi_{a}^{(0)}(x)+\beta\psi_{a}^{(1)}(x)+\beta^2\psi_{a}^{(2)}(x)+\cdots$$

such that, for every $n$, the limit of $\psi_{a}^{(n)}(x)$ as $a\to 0$ exists when $x\neq 0$, and the equation denoted by is satisfied order by order in $\beta$. This conjecture would provide numerical evidence that the limits $a\to 0$ and $\beta\to 0$ can be commuted in the solution of the Schrödinger equation; the source does not establish whether the claim is resolved.