Higher-order monotonicity conjecture for contiguous parabolic-cylinder-function ratios

Let $U(n,x)$ denote the parabolic cylinder function. For $n>1/2$, define

$$R_n^{[1]}(x)=\frac{U(n-1,x)}{U(n,x)},$$

and recursively set

$$R_n^{[k+1]}(x)=\frac{R_n^{[k]}(x)}{R_{n+1}^{[k]}(x)}.$$

Higher-order ratio monotonicity conjecture. The functions $R_n^{[k]}(x)$ are positive and increasing in $x$, satisfy $R_n^{[k+1]}(x)>R_n^{[k]}(x)$, and obey $R_n^{[k]}(x)<1$ when $k\geq 2$.

This conjecture generalizes the known monotonicity of the double ratio $W_n(x)$ and would provide a systematic hierarchy of monotonicity properties for contiguous ratios of parabolic cylinder functions. The source gives no resolution of the conjecture.