An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a positive integer $s>1$.

Let $\mathcal{O}_{n,s}=\{\lambda\vdash n: \lambda_i\in\{1,3,5,\dotsc,2s-1\}\}$. Parts are odd integers.

Also, let $\mathcal{A}_{n,s}=\{a_1+2a_2+2a_3+\dotsb+2a_s=n: a_1\geq a_2\geq a_3\geq\dotsb\geq a_s\geq0\}\subset\mathbb{Z}^s_{\geq0}$. In my earlier MO post Seeking a bijective proof enumerating two partition sets: Part I, I proposed a question on $\#\mathcal{O}_{n,s}=\#\mathcal{A}_{n,s}$ which was subsequently answered by Per Alexandersson.

Let's add one more set of partitions $$\mathcal{B}_{n,s}=\{a_1+a_2+\cdots+a_s=n: a_2\geq 2a_1, 2a_3\geq 3a_2, 3a_4\geq 4a_3,\dotsc,(s-1)a_s\geq sa_{s-1}\}.$$ I would like ask:

QUESTION. Is there a combinatorial or bijective proof of the equinumerosity $\#\mathcal{O}_{n,s}=\#\mathcal{B}_{n,s}$?