nt.number theory - An inequality relating the factorial to the primorial.

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

$$ text{Comp}[{leftlfloor n /2 rightrfloor}, n] < text{Fact}[1, {leftlfloor n /2 rightrfloor}] text{Prim}[{leftlfloor n /2 rightrfloor}, n] ? $$

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.