Sums of three cubes over a function field

We use a function field version of the circle method to prove that a positive proportion of elements in Fq[t] are representable as a sum of three cubes of minimal degree from Fq[t], assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet L-functions is known for large fixed q, via recent developments in homological stability.