Pure reversed-prime circle-method asymptotic

Let $b\geq 2$ and $k\geq 2$ be fixed. Define

$$\mathcal{R}_{0,k}(N)=\sum_{\substack{\overleftarrow{p_1},\ldots,\overleftarrow{p_k}\leq N\overleftarrow{p_1}+\cdots+\overleftarrow{p_k}=N\\(\overleftarrow{p_1}\cdots\overleftarrow{p_k},b^3-b)=1}}(\log p_1)\cdots(\log p_k)$$

and

$$\mathcal{S}_{0,k}(N):=\sum_{\substack{n_1,\ldots,n_k\leq N\ _1+\cdots+n_k=N\\(\overleftarrow{n_1}\cdots\overleftarrow{n_k},b)=1}}1.$$

Let

$$\mathfrak{S}_k(N):=\prod_{\substack{p\mid b^3-b\p\mid N}}\left(1-\left(\frac{-1}{p-1}\right)^{k-1}\right)\prod_{\substack{p\mid b^3-b\p\nmid N}}\left(1-\left(\frac{-1}{p-1}\right)^k\right).$$

Pure reversed-prime asymptotic conjecture. For every $A>0$,

$$\mathcal{R}_{0,k}(N)=\mathfrak{S}_k(N)\mathcal{S}_{0,k}(N)+O_{b,A}\left(\frac{N^{k-1}}{(\log N)^A}\right).$$

This is a major-arc circle-method prediction for sums of $k$ reversed primes. The source identifies minor-arc estimates as the key barrier; the binary case is especially out of reach.