Having failed to install on desktop with W10 Pro, I've now also failed to install on my Surface Pro 4 machine (basically a small laptop).
Using Edge as browser, after downloading the Downloads tab says
"The signature of this file is corrupt or invalid"
Trying to run it causes Windows Defender Smartscreen to throw up a blue window saying it's prevented the file from running, but gives an option to override & run it anyway. Selecting that option allows the file to run but at the end of what looks like a normal intallation process, it just gives the error message 'Setup failed' (as in screenshot in previous post).
It has of course uninstalled the previous version of Publisher so it's completely vanished from both my computer.