Raf Cluckers' conjecture on parametric suprema of constructible functions

Let $X\subset(0,\infty)\times\mathbb{R}^k$ be a subanalytic set, and let $F\colon X\to\mathbb{R}$ be a constructible, non-negative function, written $F(\varepsilon,\bm{x})$. A function is constructible when it belongs to the class generated by globally subanalytic functions and logarithms of positive globally subanalytic functions. Raf Cluckers' conjecture. There exist a constant $\delta>0$ and a function $G\colon(0,\infty)\to\mathbb{R}$ of the form

$$G(\varepsilon)=c\lvert\log\varepsilon\rvert^l\varepsilon^a$$

where $c\geq0$ is real, $a$ is rational, and $l\geq0$ is an integer, such that

$$\delta G(\varepsilon)\leq\sup_{\bm{x}\in\mathbb{R}^k}F(\varepsilon,\bm{x})\leq G(\varepsilon)$$

for all $\varepsilon<\delta$. This conjecture proposes a weak stability property for parametric maxima or suprema of constructible functions, whose full stability under parametric maxima is known to fail. The source gives no resolution, so the conjecture remains open.