The picture above does NOT represent the Hope-X vehicle as labeled, but the HSFD sub-scale demonstrator.
Here is a summary of this very interesting program, and some links to it:
Japan's Missions for RSTS Development
NAL (National Aerospace Laboratory of Japan) and NASDA (National Space Development Agency of Japan), which are two bodies among three organizations which composed JAXA, have successfully conducted four experiments to develop technologies for reusable space vehicles: OREX, HYFLEX, ALFLEX, and HSFD. The outcomes of these experiments have contributed to Japan's space technological expertise.
Japan's missions to complete
A recent biography of Burt Rutan also states that Scaled Composites worked on a "Fuji Mini Shuttle POC" vehicle at about the same time as the X-38 CRV. The ALFLEX fits very well in the "mini-shuttle" category, which is also consistent with the time sequence, but I couldn't find any further confirmation whether Rutan worked on ALFLEX or on HSFD-I, which was however a little bigger and a slightly later design.