With this Roundcube plugin you can access your Browsers Bookmarks from within Roundcube. You can use the Browser Plugin/WebExtension to automatically sync your bookmarks, every time you create, edit or remove a bookmark in your Browser. I have created a extension for Firefox, released at Firefox AddOns, a Microsoft Edge version and a Chromium version for all other Chromium based browsers, available from the Release Pages, which is not officially in the Chrome Store, since a fee of 5$ is needed to publish a free exentsion.
Browser Support This plugins works also with Bookmark Export's from Firefox, Internet Explorer and Google Chrome Browser as html file. In both browsers, the exported bookmark file seems to be compatible with this plugin. However, since I didn't use Internet Explorer or Chrome, I can't tell you if there is a way to export these bookmarks automatically. Maybe there are scripts, add-ons or advanced settings. The only reasonably comfortable way I found is in Firefox. This is described in detail in the FAQ.
Please create issues in the Codeberg.org issue tracker. I have disabled the issue tracker in GitHub.com because the repo there is only used as a workaround and for compatibility reasons.