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A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle in $G$. In other words, each cycle of $G$ must be colored with at least three colors. Given a list assignment $L=\left\{L\right(v):v\in V\}$, if there exists an acyclic coloring $\pi$ of $G$ such that $\pi \left(v\right)\in L\left(v\right)$ for all $v\in V$, then we say that $G$ is acyclically $L$-colorable. If $G$ is acyclically $L$-colorable for any list assignment $L$ with $\left|L\right(v\left)\right|\ge k$ for all $v\in V$, then $G$ is acyclically $k$-choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles...