  We use a function field version of the circle method to prove that a positive
proportion of elements in $\mathbb{F}_q[t]$ are representable as a sum of three
cubes of minimal degree from $\mathbb{F}_q[t]$, 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.
