The conjecture on infinite essential dimension of elliptic curves

Let $E$ be an elliptic curve over a number field. Its essential dimension is denoted by $\operatorname{ed}E$. Infinite essential-dimension conjecture for elliptic curves. Then

$$\operatorname{ed}E=+\infty.$$

The source proves infinite essential dimension for elliptic curves with semistable bad reduction at some prime of the number field, and suggests this statement as a broader guess for all elliptic curves over number fields.