Silverman's transcendence conjecture for canonical heights

Let $f(z)\in\overline{\mathbb{Q}}(z)$ with $d:=\deg(f)\geq 2$. For every $a\in\mathbb{P}^1(\overline{\mathbb{Q}})$, the canonical height $\hat{h}_f(a)$ is defined, and its value is either $0$ or transcendental.

Silverman's conjecture. For every $a\in\mathbb{P}^1(\overline{\mathbb{Q}})$, $\hat{h}_f(a)$ is either $0$ or transcendental.

This conjecture concerns the exact values of canonical heights in arithmetic dynamics. It is known when $f$ is linearly conjugate to $z^d$ or to $\pm C_d(z)$, where $C_d(z)$ is the Chebyshev polynomial of degree $d$, but remains open in general; in particular, it is not known whether any rational value of $a$ can yield an irrational canonical height for $f(z)=z^2+\frac{1}{2}$.