The 3-uniform partition relation at the first uncountable ordinal

Let $\mathfrak{c}$ denote the cardinality of the continuum, and let $\omega_1 = \aleph_1$ denote the first uncountable ordinal. For each integer $n \geq 2$, consider the 3-uniform ordinal Ramsey property $\mathfrak{c} \to (\omega_1, n)^3_2$, defined as follows. For any 2-coloring of all 3-element subsets of the ordinal type of $\mathfrak{c}$ (where the coloring is given as a predicate on ordered triples of distinct elements that is invariant under permutation of its arguments), either there exists a red-monochromatic subset of order type $\omega_1$ (every 3-element subset of which is colored red), or there exists a blue-monochromatic subset of cardinality $n$ (every 3-element subset of which is colored blue). Determine whether $\mathfrak{c} \to (\omega_1, n)^3_2$ holds for all integers $n \geq 2$.