Wilkie's conjecture on integer-valued definable functions

Let $\mathbb{R}_{\mathrm{an},\exp}$ denote the real field expanded by restricted analytic functions and the exponential function. Let $f:\mathbb{R}\to\mathbb{R}$ be definable in $\mathbb{R}_{\mathrm{an},\exp}$, assume that $f(a)\in\mathbb{Z}$ for all sufficiently large positive integers $a$, and suppose there is an $r>0$ such that $|f(x)|<\exp(rx)$ for all sufficiently large $x$. Wilkie's conjecture. There exist a polynomial $P(x,y_1,\ldots,y_m)$ with rational coefficients and positive real algebraic integers $\alpha_1,\ldots,\alpha_m$ such that

$$f(x)=P(x,\alpha_1^x,\ldots,\alpha_m^x)$$

for all sufficiently large $x$. This conjecture proposes an arithmetic description of definable functions in $\mathbb{R}_{\mathrm{an},\exp}$ that take integer values at all sufficiently large positive integers under an exponential growth bound. The paper draws attention to it as a broad conjecture; its resolution is not established in the supplied text.