Contour-integral bound conjecture on a V-shaped domain

Let $T$ be the truncation parameter, let $\beta\in[0,2)$ be fixed, and let $z=re^{\pm i\beta\pi/2}$ with $r\in[e^{4+2\beta-2T},1]$. Let $f(u,x)$ and $\delta(u)$ be the functions defined in the paper, and let $\Gamma$ be the positively oriented rectangle

$$[1-\beta/2,4T^2+1-\beta/2]\times[-ai,ai],\qquad a=2\pi\left(T+\frac12\log r\right).$$

Contour-integral bound conjecture. The remaining contour integral satisfies

$$\left\lvert\int_\Gamma f(u,x)\delta(u)\,du\right\rvert=\mathcal{O}(e^{-T}).$$

The claim is presented as being supported by strong numerical evidence and would extend the corresponding contour bound to the V-shaped domain; the supplied text does not provide a proof.