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Answer to Version 1 of the question:
No, let $b_1,b_2,a_1,a_2$ be $2,6,3,4$. Then $$ (1-1/2)(1-1/6)=5/12 \not\ge 1/2=(1-1/3)(1-1/4) $$$$ (1-1/2)(1-1/6)=5/12 \not\ge 1/2=(1-1/3)(1-1/4). $$
No, let $b_1,b_2,a_1,a_2$ be $2,6,3,4$. Then $$ (1-1/2)(1-1/6)=5/12 \not\ge 1/2=(1-1/3)(1-1/4) $$
No, let $b_1,b_2,a_1,a_2$ be $2,6,3,4$. Then $$ (1-1/2)(1-1/6)=5/12 \not\ge 1/2=(1-1/3)(1-1/4). $$
