Large-parameter positivity conjecture for the transmission eigenvalue variation

Consider the transmission problem for the operator $\mathcal L^{T}({\mathfrak j},\kappa)=-\frac{d^2}{dx^2}+i{\mathfrak j}x$ on the line, with non-negative $\kappa$, and set $y=\kappa {\mathfrak j}^{-1/3}$. Let $\lambda_1^T(y)$ be the unique continuous solution of

$$2\pi \operatorname{Ai}'(e^{i2\pi/3}\lambda)\,\operatorname{Ai}'(e^{-i2\pi/3}\lambda)=-y$$

satisfying $\lambda_1^T(0)=|a_1'|e^{i\pi/3}$, and define

$$\delta^T(y)=2-y\,\frac{\operatorname{Re}(\lambda_1^T)'(y)}{\operatorname{Re}\lambda_1^T(y)}.$$

Large-parameter positivity conjecture. There exists $y_1>0$ such that $\delta^T(y)>0$ on $[y_1,+\infty)$. The conjecture concerns the eventual monotonicity of the real part of the first transmission eigenvalue under semiclassical scaling; the preceding lemma establishes the analogous positivity on an interval beginning at $0$, but no proof of the large-$y$ assertion is provided here.