Monotonicity of the optimal stopping region for the duration problem

Let $U_n(x)$ be the expected duration of owning the relatively best object whose rank remains within the two when the time to go is $n$, conditional on accepting the relatively best applicant with value $x$. Define

$$G_n(x)=U_n(x)-\sum_{k=1}^{n-1}x^{k-1}\int_x^1U_{n-k}(y)\,dy.$$

The stopping region is $B=\{(n,x):G_n(x)\geq 0\}$. Monotonicity conjecture. If $G_n(x)\geq 0$, then $G_n(y)\geq 0$ for every $y\geq x$; equivalently, the optimal stopping rule is a threshold rule in the observed maximum. This would establish optimality of the 1-SLA stopping rule for the full-information duration problem when stopping is restricted to relatively best objects. The analogous implication $G_n(x)\geq 0\Rightarrow G_{n-k}(x)\geq 0$ is stated to have been proved, while this monotonicity assertion remains open.