The answer is no. E.g., if $p_1=1/2$, $p_2=1/2$, $q_1=1/100$, $q_2=99/100$, and $\beta=1/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $0.00877\ldots<1$.

Jiacai Liu asked if the conjectured inequality holds in the opposite direction. The answer to this is also no. E.g., if $p_1=1/1000$, $p_2=999/1000$, $q_1=9/10$, $q_2=1/10$, and $\beta=6/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $1.5006\ldots>1$.

The answer is no. E.g., if $p_1=1/2$, $p_2=1/2$, $q_1=1/100$, $q_2=99/100$, and $\beta=1/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $0.00877\ldots<1$.

Jiacai Liu asked if the conjectured inequality holds in the opposite direction. The answer to this is also no. E.g., if $p_1=1/1000$, $p_2=999/1000$, $q_1=9/10$, $q_2=1/10$, and $\beta=6/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $1.5006\ldots>1$.

The answer is no. E.g., if $p_1=1/2$, $p_2=1/2$, $q_1=1/100$, $q_2=99/100$, and $\beta=1/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $0.00877\ldots<1$.

The answer is no. E.g., if $p_1=1/2$, $p_2=1/2$, $q_1=1/100$, $q_2=99/100$, and $\beta=1/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $0.00877\ldots<1$.

Jiacai Liu asked if the conjectured inequality holds in the opposite direction. The answer to this is also no. E.g., if $p_1=1/1000$, $p_2=999/1000$, $q_1=9/10$, $q_2=1/10$, and $\beta=6/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $1.5006\ldots>1$.