France Pathé Blu-ray, 4k resto. It has both original negative image cut (not tinted) and a first gen tinted cut owned by BFI but not English-friendly

Japan has 4k restoration in Blu-ray but no EN subs

Janus acquired the rights so hopefully a Criterion release some day

Grapevine Blu-ray

The latest and best restoration from official screencaps are from Shout Factory Silent Film Festival + Cinémathèque Française https://silentfilm.org/preservation/the-italian-straw-hat/ which is from OCN and original France title negative. Janus seems to have acquired the rights https://www.janusfilms.com/films/2259 The Grapevine Blu-ray (including their updated 2K transfer) or the Flicker Alley DVD might be the ones referred in the above Shout Factory Silent Film Festival as "US viewers have had to be satisfied with either the alternate version that was released in the United States in 1931, or hybrids that supplement the US version with additional footage"

Japan and France has 4k restoration in Blu-ray but no EN subs

France and Japan has 4k restoration in Blu-ray but no EN subs