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.