I am using Chrome to connect from Windows 10 1903 in my tests.
When I print I get a pdf file with a fixed name in the Downloads folder.
If if want to print it I have to click on the downloaded file to do so, even though I clearly stated my intent to print in the session.
While it is possible to print this way, it is not an easy sell to users.
A common way to deal with similar problems in web applications is to pop a window with the PDF content. I can then select print or download as pdf from that window as I wish.
The popped window would make the experience for a user a lot better and should not be a large effort to implement.
Do you have any plans along these lines?