Wilkie's conjecture on rational points in the exponential real field

Let $X\subseteq\mathbb{R}^n$ be definable in the o-minimal structure $\mathbb{R}_{\exp}$, and let $X^{\mathrm{alg}}$ be the union of all semi-algebraic curves contained in $X$. Define

$$X^{\mathrm{trans}}=X\setminus X^{\mathrm{alg}}.$$

For $q=a/b\in\mathbb{Q}$ with $\gcd(a,b)=1$, let $H(q)=\max(|a|,|b|)$, extend $H$ to tuples by taking the maximum of the coordinate heights, and set

$$X(\mathbb{Q},H)=\{x\in X\cap\mathbb{Q}^n\mid H(x)\leq H\}.$$

Wilkie's conjecture. There exist constants $c=c(X)$ and $d=d(X)$ such that, for every $H\in\mathbb{N}$ with $H>e$,

$$\left|X^{\mathrm{trans}}(\mathbb{Q},H)\right|\leq c\log(H)^d.$$

This conjecture predicts a polylogarithmic refinement of the Pila--Wilkie counting bound for sets definable in $\mathbb{R}_{\exp}$, replacing the general bound $cH^\epsilon$. It is presented as conjectural in the source, and no resolution is supplied here.