Equivalence of amalgamation, type amalgamation, coloring, and pattern conditions

Let $T$ be a simple theory and let $3 \leq k \leq \aleph_0$. Consider the following conditions: (A) $T$ has $\mathcal{P}^-(k)$-amalgamation of models; (B) $T$ has $<k$-type amalgamation; (C) for every regular uncountable $\theta$, every $\mu=\mu^{<\theta}$, and every $\lambda\leq 2^\mu$, $T$ has the $(\lambda,\mu,\theta,k)$-coloring property, and this remains true in every forcing extension; and (D) $T$ does not admit $\Delta_{k'+1,k'}$ for any $k'<k$. Equivalence conjecture. Conditions (A), (B), (C), and (D) are equivalent. The result would identify model amalgamation, type amalgamation, coloring properties, and forbidden patterns as equivalent dividing lines for simple theories; the preceding corollary establishes only the implication chain (A) $\Rightarrow$ (B) $\Rightarrow$ (C) $\Rightarrow$ (D), with the middle implication cited as known.