Theorem 10.21 (see page 268) of the book "Chromatic Graph Theory" (which is written by Gary Chartrand and Ping Zhang), confirm that this conjecture is proved and in $2001$, Robertson, Saunders, Seymour and Thomas announced they verified this conjecture.
This conjecture is known as $4$-flow conjecture, and Seymour showed that the $4$-flow conjecture is equivalent to the following more general conjecture:
Each bridgeless matroid without $F_7^*$, $M^*(K_5)$, or $M(P_{10})$ minor has a nowhere-zero flow over $GF(4)$, where $P_{10}$ shows the Peterson graph.
For cubic graphs, this conjecture is verified in the paper "Tutte’s Edge-Colouring Conjecture", which is written by the above authors. Also, note that most conjectures about flows can be easily reduced to the case of cubic graphs by splitting arguments.
Anyway, I could not find any paper which states the proof of this conjecture. Maybe, you can ask to someone which know the authors or email directly to one of these people. It seems that the conjecture is verified since it is stated in famous book as theorem.
The following link maybe useful: $4$-flow conjecture