  We find a lower bound for the Hausdorff dimension that a Liouville Brownian
motion spends in $\alpha$-thick points of the Gaussian Free Field, where
$\alpha$ is not necessarily equal to the parameter used in the construction of
the geometry. This completes a conjecture in \cite{berestycki2013diffusion},
where the corresponding upper bound was shown.
  In the course of the proof, we obtain estimates on the (Euclidean)
diffusivity exponent, which depends strongly on the nature of the starting
point. For a Liouville typical point, it is $1/(2 - \frac{\gamma^2}{2})$. In
particular, for $\gamma > \sqrt{2}$, the path is Lebesgue-almost everywhere
differentiable, almost surely. This provides a detailed description of the
multifractal nature of Liouville Brownian motion.
