Silverman's arithmetic-order trichotomy conjecture

Let $X/K$ be a smooth projective variety over a number field, let $H$ be a Weil height associated to an ample divisor, and define the arithmetic order $\mathsf{A}(X)$ using the iterated-logarithm height-counting limit in the source. Silverman's arithmetic-order trichotomy conjecture. The arithmetic order $\mathsf{A}(X)$ exists and satisfies

$$\mathsf{A}(X)=0\quad\text{or}\quad\mathsf{A}(X)=1\quad\text{or}\quad\mathsf{A}(X)=\infty.$$

Equivalently, height counting has, up to lower-order terms, one of three coarse regimes: power growth in $T$, power growth in $\log T$, or bounded growth. The source attributes this conjecture to Silverman and gives no resolution.