Five in Seven

I believe this pentagon is the largest which can be inscribed in a heptagon. I found it numerically. It touches the heptagon in four places. It is symmetric about a central axis, $OD$ . $GA \parallel IH$ . It is inclined at $20 \frac{4}{7}^\circ$ with respect to the heptagon. If someone can prove this is the largest pentagon possible or provide a larger pentagon, I would be most interested. Thanks!

If the side of the heptagon is one, then $IH \approx 1.2359303924$ and $S \approx 2.628070408$ .