It is given in the book by El Gamal that

The capacity region of the DM-BC depends on the channel conditional pmf $ p(y_1 , y _2 |x)$ only through the conditional marginal pmfs $ p(y_1 |x)$ and $ p(y_ 2 |x)$

The statement made is about the entire capacity region. Every point in the capacity region means that there exists a sequence of $ (2^{nR_1},2^{nR_2},n)$ (considering only private messages) codes with 0 error. I am wondering about the converse.

For a given sequence of codes, would broadcast channels with identical marginals have the same achievability, ie, if it is achievable in the first channel, would it be achievable in the second channel and vice versa?