  The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$, introduced by Bergeron and
Pr{\'e}ville-Ratelle, are defined as a poset of $m$-Dyck paths equipped with
the generalized rotation order, and constitute a Fuss-Catalan generalization of
the classical Tamari lattices $\mathcal{T}_{n}$. While for $\mathcal{T}_{n}$
many combinatorial realizations are known, to present there is no further
combinatorial realization of $\mathcal{T}_{n}^{(m)}$. In this article, we
introduce a certain decomposition of $m$-Dyck paths into $m$-tuples of Dyck
paths, and after a certain modification of these $m$-tuples, we conjecture that
the resulting $m$-tuples of Dyck paths realize $\mathcal{T}_{n}^{(m)}$ as an
induced subposet of the $m$-fold direct product of $\mathcal{T}_{n}$ with
itself. We are able to prove this conjecture for $n\leq 3$, and provide
necessary conditions for $m$-tuples of Dyck paths to belong to this
realization. However, for $n\geq 5$, no sufficient condition is known.
