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 $P_{10}$ denotes 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

$\textbf{Added later}:$ At the page 183 of the book "Topics in Chromatic Graph Theory" by Lowell W. Beineke and Robin J. Wilson (published in 2015), it is stated that (Conjecture B) this conjecture is still open.

$\textbf{Added later}(6/21/2017):$ The valuable book "Graph Theory: 5th edition" by Reinhard Diestel, published in 2017 and I prepared a copy of this book. In part 6.6 of this book (Tutte’s ﬂow conjectures), there is a detailed information about $4$-flow conjecture. In this part, it is stated that this conjecture is still open. Some part of this book:

$\ldots$ Even if true, the $4$-ﬂow conjecture will not be best possible: a $k^{11}$, for example, contains the Petersen graph as a minor but has a $4$-ﬂow,
even a $2$-ﬂow. The conjecture appears more natural for sparser graphs; a proof for cubic graphs was announced in 1998 by Robertson, Sanders, Seymour and Thomas $\ldots$.