Does $2^\lambda\to(\kappa_\alpha+1)^{r+1}$ imply $\lambda\to(\kappa_\alpha)^r$? (Erdős #1167)

Let $r\geq 2$ be a finite integer, let $\lambda$ be an infinite cardinal, and let $\gamma$ be an ordinal with an infinite cardinal $\kappa_\alpha$ assigned to each $\alpha<\gamma$. Using the standard partition-arrow notation, is it always true that $$2^\lambda\to(\kappa_\alpha+1)_{\alpha<\gamma}^{\,r+1}$$ implies $$\lambda\to(\kappa_\alpha)_{\alpha<\gamma}^{\,r}\,?$$ Here $+$ is cardinal addition, so $\kappa_\alpha+1=\kappa_\alpha$ whenever $\kappa_\alpha$ is infinite. (The arrow $\mu\to(\mu_\alpha)_{\alpha<\gamma}^{s}$ means: for every colouring of the $s$-element subsets of a set of size $\mu$ by colours indexed by $\alpha<\gamma$, some colour $\alpha$ has a homogeneous subset of size $\mu_\alpha$.)

Acceptance. FULLY RESOLVES: a complete proof, in ZFC, that $2^\lambda\to(\kappa_\alpha+1)^{r+1}$ implies $\lambda\to(\kappa_\alpha)^r$ for all admissible parameters (machine-checkable in Lean/Coq preferred, given the formalised statement), OR a counterexample — explicit parameters $\lambda,r,\gamma,(\kappa_\alpha)$ together with a model of set theory (e.g. via forcing, with the consistency strength stated) in which the hypothesis holds but the conclusion fails, with proof. ADVANCES: a proof of the implication for a genuinely new parameter regime not previously known (for instance a fixed small $r$, or specified cardinal patterns), with full proof; or a consistency result establishing independence for some parameter regime. Deliver the written or formalised proof, or the forcing construction with verification of the required relations.