Let $ \pi_{n}(x) $ denote the number of prime gaps of size $ n $ below $ x $ for even $ n $. The Hardy-Littlewood conjecture predicts that $ \pi_{n}(x)\sim C_{n}\frac{x}{\log^{2}x} $ with $ C_{n}=C_{2}\prod_{q\mid n}\frac{(q-1)}{(q-2)} $, $ C_{2}=0.66016... $ is the so called twin prime constant and $ q $ runs among odd primes dividing $ n $. Let $ Q_{a,b}(x) $ be the largest primorial not exceeding $ x^{a}\log^{b} x $.
Assuming the conjecture above, can we determine $ a $ and $ b $ so that $ \pi_{Q_{a,b}(x)}(x)\asymp 1 $ as $ x $ tends to $ \infty $?