Asymptotic distribution conjecture for pure imaginary zeros of modified Bessel functions

Let $K_{\alpha}(x)$ be the modified Bessel function of the second kind and let $K'_{\alpha}(x)$ denote its derivative with respect to $x$, with $x>0$. Consider the pure imaginary zeros $i\nu_n$ of the combined conditions involving $K_{\alpha}(x)$ and $K'_{\alpha}(x)$, indexed by $n$ for $n\gg1$.

Asymptotic distribution conjecture. The asymptotic location of the $n$-th pure imaginary zero is determined by

$$\nu_n\,\operatorname{arccosh}\biggl(\frac{\nu_n}{x}\biggr)-\sqrt{\nu_n^2-x^2}=\frac{(n+\tfrac12)\pi}{2}.$$

The $n$-th pure imaginary zero of $K_{\alpha}(x)$ alone or of $K'_{\alpha}(x)$ alone is expected to obey the same rule, with $n$ on the right-hand side replaced by $2n$. This is a WKB/Bohr–Sommerfeld prediction for the large-index distribution of these zeros; the source presents it as a conjecture rather than as a proved asymptotic theorem.