Limit points of $ \alpha $ -normalized prime gaps

Question

Let $\mathbb{L}_{\alpha} $ be the set of limit points of $ \{\dfrac{p_{n+1}-p_{n}}{\log^{\alpha}p_n}\}_{n=1..\infty} $. Erdös conjectured that $\mathbb{L}_{1}=[0,\infty] $. Has it been conjectured that $ \lambda(\mathbb{L}_{1-x})\sim\lambda(\mathbb{L}_{1+x}) $ for all $ x\in[0,1) $ where $ \lambda $ is the Lebesgue measure? If yes do we know whether this quantity is a decreasing function of $ x $ ?