The ordinal partition relation for triples at the continuum with double omega and four

Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha, \beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for any 2-coloring of all 3-element subsets of the ordinal type $\alpha$ (where the coloring is given by a symmetric predicate on ordered triples of distinct elements), either there exists a subset $s \subseteq \alpha$ of order type $\beta$ such that every 3-element subset of $s$ is colored red, or there exists a subset $s \subseteq \alpha$ of cardinality $c$ such that every 3-element subset of $s$ is colored blue. Let $\omega$ denote the first infinite ordinal. Does the partition relation $\mathfrak{c} \to (\omega \cdot 2, 4)^3_2$ hold?