Recently I have asked two questions in MSE regarding the simultaneous solutions of the following inequalities: $$x+y>p_{\pi(x)}+p_{\pi(y)+1}\tag{1}$$$$x+y>p_{\pi(y)}+p_{\pi(x)+1}\tag{2}.$$ My earlier question regarding this inequality was answered in MSE. My next question is the following:

Let $x+y+2$ be a prime such that $x>y\ge 3$ and $x$ and $y$ both are composites. Is it true that if $(1)$ holds, then $(2)$ must hold? Are there infinitely many counterexamples to this assertion?

Can anyone tell me whether this would be a suitable question for MO?