Short Rainbow Circuits in Regular Matroids

DeVos et al conjectured that if $M$ is a simple, regular matroid and $c$ is a colouring of the elements of $M$ with $r(M)+1$ colours, where each colour class has at least two elements, then $M$ contains a rainbow circuit of size at most $\lceil \frac{r(M)+1}{2}\rceil \mathrm{.}$ We prove this conjecture by showing that for all such regular matroids there are four rainbow circuits $C_i,i=1,2,3,4$ for which ${\sum }_i\mathrm{\mid }{C}_i\mathrm{\mid }\le 2r(M)+4$ and for which no element of $M$ belongs to more than two of the circuits.