Let [a,b] = {k integer | a < k <= b}. Further let

- Comp[a,b] = product_{c in [a,b]} c composite;
- Fact[a,b] = product_{k in [a,b]} k integer;
- Prim[a,b] = product_{p in [a,b]} p prime.

*Question*: For n > 2 and n not in {10,15,27,39} is it true that

$$ \text{Comp}[{\left\lfloor n /2 \right\rfloor}, n] < \text{Fact}[1, {\left\lfloor n /2 \right\rfloor}] \ \text{Prim}[{\left\lfloor n /2 \right\rfloor}, n] \ ? $$

*Update*: The state of affairs: Gjergji Zaimi showed that for large enough n the inequality is true. In my answer I affirm that the inequality is true in the range 40 <= n <= 10^5. It remains open whether 10^5 is 'large enough' in the sense of Gjergji's analysis.