A Steiner triple system which colors all cubic graphs

We prove that there is a Steiner triple system T such that every simple cubic graph can have its edges coloured by points of T in such a way that for each vertex the colours of the three incident edges form a triple in T . This result complements the result of Holroyd and Škoviera that every bridgeless cubic graph admits a similar colouring by any Steiner triple system of order greater than 3. The Steiner triple system employed in our proof has order 381 and is probably not the smallest possible.