How to convert 16:9 to 4:3 and then back to 16:9 without losing part of the image?

Question

Iron

There is a rather non-trivial problem. There is a PC with an aspect ratio of 16 * 9, there is an LED screen with the same aspect ratio. And there is a card that converts PC output to LED screen signal, but its target ratio is 4:3. As a result, after the output, part of the image from the monitor is cut off (the bar below). It's expensive to buy a new board, and the solution doesn't come to mind right away. Maybe there are some programs for adapting the output image? Maybe mathematically you can calculate the screen resolution for the correct output?
In general, I'm in a stupor. I would be glad for any tips and help.