  Let $G$ be a bridgeless cubic graph. In 2023, the three authors solved a
conjecture (also known as the $S_4$-Conjecture) made by Mazzuoccolo in 2013:
there exist two perfect matchings of $G$ such that the complement of their
union is a bipartite subgraph of $G$. They actually show that given any
$1^+$-factor $F$ (a spanning subgraph of $G$ such that its vertices have degree
at least 1) and an arbitrary edge $e$ of $G$, there exists a perfect matching
$M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. This
is a step closer to comprehend better the Fan--Raspaud Conjecture and
eventually the Berge--Fulkerson Conjecture. The $S_4$-Conjecture, now a
theorem, is also the weakest assertion in a series of three conjectures made by
Mazzuoccolo in 2013, with the next stronger statement being: there exist two
perfect matchings of $G$ such that the complement of their union is an acyclic
subgraph of $G$. Unfortunately, this conjecture is not true: Jin, Steffen, and
Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts.
Here we show that, despite of this, every cyclically 3-edge-connected cubic
graph satisfies this second conjecture.
