In TruClient 12.57, Firefox version is 40, Chromium version is 63, TruClient Browser version is 55 (equivalent to Firefox 55). TruClient IE is based on same core as local IE. That means, if IE 11 is installed, TruClient IE is based on IE 11 core.
As far as I know, by now, there is no plan to run TruClient on local browsers (for local browsers, I guess you mean "real browsers" installed in local OS, such as real IE, real Firefox and real Chrome). The major reason is TruClient needs some browser features to drive browsers to test various areas, but these features are not supported by local browsers. TruClient has to use the modified versions.
TruClient has a lite version, TruClient Lite, which runs on local Chrome. It's available at https://chrome.google.com/webstore/detail/truclient-lite-for-chrome/onkiafmnfodnfigdpdoaeafjogedfcpo?hl=en.