Take some 4 consecutive primes $p_n,p_{n+1},p_{n+2},p_{n+3}$ where $p_n \geq 5$.

Now form two products: $p_n \cdot p_{n+3}$ and $p_{n+1} \cdot p_{n+2}$.

Is there always at least one prime in the interval $[p_n \cdot p_{n+3},p_{n+1} \cdot p_{n+2}]$ (it does not matter if we have that $p_n \cdot p_{n+3}>p_{n+1} \cdot p_{n+2}$, if that is the case then just reverse the endpoints of the interval)?