Let $ a $ and $ b $ be two positive integers such that $ a\lt b $ and $ ab $ is a primorial. Let $\mathcal{N}(x)=\mathcal{N}_{prime}(x)+\mathcal{N}_{pure}(x)+\mathcal{N}_{mixed}(x)$ where $ \mathcal{N}(x) $ is the number of such pairs $ (a,b) $ with $ b\le x $, $\mathcal{N}_{prime} (x)$ the number of such pairs $ (a,b) $ with $ b\le x$ fulfilling the additional requirement $ (b-a,b+a)\in\mathbb{P}^{2} $, $ \mathcal{N}_{pure}(x) $ defined similarly but with $\Lambda(b-a)\Lambda(a+b)>0 $ and $ \Omega(b^2-a^2)>2 $ and finally $ \mathcal{N}_{mixed}(x) $ the number of such pairs $ (a,b) $ with $ b\le x $ such that $ \Lambda(b-a)\Lambda(a+b)=0 $ where $ \Lambda $ is the Von Mangoldt function and $ \Omega(n) $ the total number of prime factors of $ n $ counted with multiplicities.

Does one have $ \mathcal{N}_{mixed}(x)=o(\mathcal{N}_{prime}(x)) $?