First flying in early 2023, it has two more times since. It's current manifest stretches into 2033 (with a low flight-rate).
A joint effort by MHI Launch Services and JAXA, H-III was developed to provide launch capabilities for a variety of commercial satellites - whether Japanese or otherwise. The JAXA side of the work was authorized in 2013. the LE-9 engine was developed to power the first-stage of the H-III.
Launch to augment the existing X-Band communications capabilities of the Japanese military.