  Certain rigid irregular $G_2$-connections constructed by the first-named
author are related via pullbacks along a finite covering and Fourier transform
to rigid local systems on a punctured projective line. This kind of property
was first observed by Katz for hypergeometric connections and used by Sabbah
and Yu to compute irregular Hodge filtrations for hypergeometric connections.
This strategy can also be applied to the aforementioned $G_2$-connections and
we compute jumping indices and dimensions for their irregular Hodge
filtrations.
