Constantin–Strauss–Varvaruca conjecture on the periodic Hilbert transform kernel

Let $d>0$ and let $\beta_d$ be the kernel of the periodic Hilbert transform on the strip of depth $d$, defined for $s\in\mathbb{R}\setminus 2\pi\mathbb{Z}$ by the kernel formulas in the source. In particular, with $x=\pi^2/(2d)$, consider $\beta_d(\pi/2)$ as a function of $x\in(0,\infty)$. Constantin–Strauss–Varvaruca conjecture. For every $d\in(0,\infty)$,

$$\beta_d(\pi/2)\geq 1.$$

The conjecture concerns a pointwise lower bound for the periodic Hilbert transform kernel arising in the finite-depth water-wave problem. The source reports that numerical computations suggest the issue is subtle; the supplied status evidence does not establish a resolution.