Monotonicity conjecture for the alternative discrete Painlevé I solutions

Let $b_n(t)$ and $a_n(t)$ denote the positive solutions and associated quantities introduced in the paper, with $n$ a nonnegative integer. For fixed $t>0$, the expressions $a_n(t)/n$ are understood for $n\geq 1$. Monotonicity conjecture. If $0<t_1<t_2$, then

$$b_n(t_1)<b_n(t_2),\qquad a_n(t_1)>a_n(t_2),$$

so $b_n(t)$ is monotonically increasing and $a_n(t)$ is monotonically decreasing for $t>0$. For fixed $t>0$,

$$\sqrt{t}<b_n(t)<b_{n+1}(t),\qquad \frac{1}{2\sqrt{t}}>\frac{a_n(t)}{n}>\frac{a_{n+1}(t)}{n+1}>0.$$

The conjecture is suggested by numerical plots and corroborates the preceding conjecture and lemma; the monotonicity of $b_n(t)$ in $t$ is explicitly noted as analytically unproved, and no resolution is supplied here.