Complete monotonicity of squared derivative ratios of Mill's ratio

Let $r$ denote Mill's ratio, and write $r^{(n)}$ for its $n$th derivative. A function is completely monotone (CM) if all derivatives exist on $(0,\infty)$ and satisfy the alternating-sign inequalities. The Mill's-ratio derivative-ratio conjecture. For every integer $n$, the function

$$z\;\mapsto\;-\frac{\left(r^{(n)}(z)\right)^2}{r^{(2n+1)}(z)}$$

is CM. The claim refines the preceding proved result with the argument $\sqrt z$; the paper reports simulation support but no proof.