This figure describes an interference channel. 
A discrete memoryless interference channel is said to be very weak if: $$I(U_1;Y_1)\geq I(U_1;Y_2|X_2), ~~\forall ~ (U_1,X_1,X_2) \sim p(u_1,x_1)p(x_2),$$ $$I(U_2;Y_2)\geq I(U_2;Y_1|X_1), ~~\forall ~ (U_2,X_2,X_1) \sim p(u_2,x_2)p(x_1).$$
And it can be shown (see proposition 2 of here) that the maximum achievable sum-rate of the Han-Kobayashi inner bound (under very weak interference setting) is $$\text{SR}_{HK}=\max_{(R_1,R_2)\in \text{ H-K inner bound}}\big\{R_1+R_2 \big\}=\max_{(X_1,X_2) \sim p(x_1)p(x_2)} \big\{I(X_1;Y_1)+I(X_2;Y_2)\big\}.$$
Also, someone shows (see table 1) that there are cases (note that here $\lambda > 1$ strictly) such that $$\max_{(R_1,R_2)\in \text{ H-K inner bound}}\big\{ \lambda R_1+R_2 \big\} < \max_{(R_1,R_2)\in \text{Capacity region}}\big\{ \lambda R_1+R_2 \big\}.$$
That is, HK inner bound is not in general tight.
So, when we choose $\lambda=1$, does $$\underbrace{\max_{(R_1,R_2)\in \text{ H-K inner bound}}\big\{R_1+R_2 \big\}}_{\text{SR}_{HK}}=\max_{(R_1,R_2)\in \text{Capacity region}}\big\{R_1+R_2 \big\}?$$
And ideas/discussions are welcome. Also, any related suggested books/videos/slides will be perfect. Thanks.