A number is “normal” (in a given base) if every finite digit pattern occurs with the expected limiting frequency in its infinite expansion. Wolfram MathWorld’s overview of normal numbers notes that proving normality for specific famous constants is notoriously difficult, and that only specially constructed examples (such as the Champernowne constant) are known to be normal by proof.
For π specifically, authoritative discussions describe normality as conjectured rather than established. David H. Bailey’s computational work tested statistical properties of an enormous finite prefix (roughly four trillion hexadecimal digits) and found distributions consistent with normality, but such tests cannot establish the required limiting-frequency property over an infinite expansion. That gap—finite empirical evidence versus an infinite mathematical guarantee—is why π’s normality remains an open problem.