This question is a variation of a question that I asked earlier. In the earlier question, I did not include certain conditions, which then leads to some counterexamples. Head over there to view the discussion if you are interested.
Let $x_1, x_2, y_1, y_2, z_1, z_2$ be non-negative real numbers and $a_1, a_2, b_1, b_2$ be positive integers such that $$\frac{x_1}{a_1} \geq \frac{y_1}{1} \geq \frac{z_1}{b_1} \ \text{ and } \ \frac{x_2}{a_2} \geq \frac{y_2}{1} \geq \frac{z_2}{b_2}.$$ It is also given that $a_1 + b_1 = a_2 + b_2$ and $y_2 + z_2 \geq z_1 \geq z_2$.
Is the following inequality always true? $$\frac{x_1 + y_1 + x_2}{a_1 + 1 + a_2} \geq \frac{z_1 + y_2 + z_2}{b_1 + 1 + b_2}$$