Given some $\epsilon > 0$, are there infinitely many solutions $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$|\pi - \frac{a}{b}| < \frac{\epsilon}{b^2}?$$

According to https://math.osu.edu/sites/math.osu.edu/files/What_is_2018_Markov_Lagrange_Spectra.pdf

  if we prove that $\pi$ has unbounded coefficients in its continued fraction, then the answer to the above question is true.

Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$

According to this document, if we prove that $\pi$ has unbounded coefficients in its continued fraction expansion, then the answer to the above question is affirmative.