Is it true that newly open superchargers are free for about a month?
Not really. It has been true that if the supercharger is operational, but not yet listed in the car's navigation system or on Tesla's web map, then people
might get free charging sessions until the site is added and those systems updated. How long this period lasts isn't some set time, though. And this doesn't necessarily happen at all with every new station. What's more, sometimes people were retroactively charged for earlier sessions they thought had been free. Tesla has lately changed/improved their processes for updating their systems to show new locations, so it's less regular for there to be a significant wait between a supercharger being commissioned/operational and when it gets listed in Tesla's system. In fact, in some recent cases, Tesla has added the stations to their systems before the superchargers were operational at all and then just listed the location as "Temporarily Closed".
All of this to say, it sometimes happens that a newly opened supercharger
may give free charging sessions for a (usually) short period of time between activation and Tesla's system being updated, but there's no guarantee that they will or that the free charging session will stay free forever (i.e. you might get billed retroactively).