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This conjecture is equivalent to Dickson's (and Bunyakowsky's, respectively), because you can transform your instance into another which forces the compositeness you want. If you want $n+2, n+4$ to be composite, just pick some big primes (let's do 101 and 103) and constrain $n+2$ to be divisible by 101 and $n+3$ to be divisible by 103. Then use the CRT to ...

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