Shelah's eventual categoricity conjecture in tame AECs with primes

Two new cases of Shelah's eventual categoricity conjecture are established: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation and no maximal models. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}$. Assume that $K$ is $H_2$-tame and $K_{\ge H_2}$ has primes over sets of the form $M \cup \{a\}$. If $K$ is categorical in some $\lambda > H_2$, then $K$ is categorical in all $\lambda' \ge H_2$. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof is that Shelah's categoricity conjecture holds in the context of homogeneous model theory: $\mathbf{Theorem}$ Let $D$ be a homogeneous diagram in a first-order theory $T$. If $D$ is categorical in a $\lambda > |T|$, then $D$ is categorical in all $\lambda' \ge \min (\lambda, \beth_{(2^{|T|})^+})$.