Erdős Problem #1167 — Stepping Down Cardinal Partition Relations

Let $r\in\mathbb N$ with $r\ge2$, let $\lambda$ be an infinite cardinal, let $\gamma$ be an ordinal with $\gamma\ge2$, and let $(\kappa_\alpha)_{\alpha<\gamma}$ be a family of cardinals. For cardinals $\mu$ and $(\nu_\alpha)_{\alpha<\gamma}$, write

$$\mu\longrightarrow(\nu_\alpha)_{\alpha<\gamma}^{s}$$

to mean that for every set $X$ of cardinality $\mu$ and every coloring $c:[X]^s\to\gamma$ of the $s$-element subsets of $X$, there exist an index $\alpha<\gamma$ and a subset $H\subseteq X$ of cardinality $\nu_\alpha$ such that every $s$-element subset of $H$ has color $\alpha$. Here $\kappa_\alpha+1$ uses cardinal addition. Is it true that

$$2^\lambda\longrightarrow(\kappa_\alpha+1)_{\alpha<\gamma}^{r+1} \quad\Longrightarrow\quad \lambda\longrightarrow(\kappa_\alpha)_{\alpha<\gamma}^{r}?$$