Hindry–Pacheco conjecture for elliptic curves over function fields

Let $K$ be a fixed function field and let $E_i$ be a family of pairwise non-isomorphic elliptic curves over $K$. For each $E_i$, let $c_{E_i/K}$ be defined by the leading term of its $L$-function at $s=1$, and let $h(E_i)$ be its logarithmic height. Hindry–Pacheco conjecture. One has

$$\lim_{i\to\infty}\frac{\log |c_{E_i/K}|}{h(E_i)}=0.$$

This is a Brauer–Siegel-type conjecture for elliptic curves over a fixed function field; the source presents it as a conjecture concerning the asymptotic size of the leading coefficient and notes that the height may equivalently be replaced, up to order of growth, by the conductor degree.