De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the family of all elliptic curves over $\mathbb{F}_q(t)$, ordering by height.

Bhargava and Shankar has already computed the average size of $n$-Selmer groups for $n=2,3,4,5$ over the family of elliptic curves over $\mathbb{Q}$, and they have formulated the conjectural average for general $n$.

Question: Is there any work so far that calculates the precise average of $n$-Selmer group of elliptic curves over $\mathbb{F}_q(t)$? Do we expect same result for the function field case, and does the geometry of number technique carry over to the function field case as well?

Thanks!