Generalization of Goldbach's ternary problem with almost equal terms

An asymptotic formula is obtained for the number of representations of a sufficiently large natural $N$ in the form $b_1p_1+b_2p_2+b_3p_3=N$ with the conditions
$$\left|b_ip_i-\frac{N}3\right|\le H, H\ge (b_1b_2b_3)^\frac43N^\frac23(\ln N)^{60}, b_i\le(\ln N)^{B_i},$$
where $b_1$, $b_2$ $b_3$, $N$ are pairwise coprime natural numbers, $B_i$ — arbitrary fixed positive numbers.