Frostman dimension of Furstenberg measure for $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ random matrix products

For compactly supported $\mu \in \mathcal{P}(\mathrm{S}\mathrm{L}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on Hölder spaces in this setting.