Parabolic cylinder function ratio inequality

Question

I would like to prove the following inequality involving the parabolic cylinder function $D_{-\nu}(x)$ for $\nu>0$:

$$ \frac{2\hspace{1pt}D_{-\nu+1}(x)}{D_{-\nu}(x)} \;<\; x+\sqrt{x^2+4\hspace{1pt}(\nu+1)} ,\quad x\ge 0 $$

The inequality seems to be tight as $x\to\infty$ (for fixed $\nu$), and as $\nu\to\infty$ (for fixed $x$).

I managed to reduce the proof of monotone likelihood ratio of the fixed input correlation coefficient distribution to this inequality.

Edit: I found a paper that actually proves this: https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.12401.