Software keyboard keeps popping back up with external keyboard attached

I have just installed a Logitech Rugged Folio keyboard on my 10th Gen iPad with iPadOS 17.3.1. The keyboard connects through the connector on the side of the iPad. The keyboard has been updated to the latest software version, according to the Logitech app.

When using the external keyboard, the screen gets covered with the on screen virtual keyboard.

Pressing the 'hide keyboard' on the virtual keyboard or the external keyboard will close it, but it will reopen as soon as I press a button on the keyboard or select any input field on the screen.

I've gone through all the settings related to the keyboard, but none of the settings have any effect on this behavior.

It renders the external keyboard completely useless because if I cannot remove it, I might as well use the virtual keyboard. How do I make the virtual keyboard disappear permanently?