@david.gray  If you haven't already done this, since Dell's documentation tends to be pretty good, I'd recommend reading through the User Guide here.  Page 68 notes that the KVM Switch function is only available in PBP mode.  I presume that's because PBP is the only time you have output from two different source devices displayed at the same time, and therefore that should be the only time you ever need to switch which of your two source devices gets to use the keyboard and mouse attached to the display.  However, when you only have a single source device occupying all of the display real estate, you can use the USB Selection function in the display's menu to associate the video input used by that source device to a particular upstream USB connection, in which case whenever that video input is the ONLY active input on the display, i.e. you are not using PBP, the display should also switch over to using the corresponding USB upstream interface.

That said, I don't understand the setup you're describing at all.  First, what application are you using to create these three virtual monitors stacked horizontally?  And am I understanding correctly that you've got each PC using three virtual monitors within its own half of the overall U4919DW display area?  So basically each PC gets its own 2560x1440 tile from the display, and then within each PC you've carved that 2560x1440 tile into three stacked tiles of 2560x480 each?  Or does "stacked horizontally" mean three tiles of 853x1440 each?  Either way, any display virtual or otherwise that is either only 480 pixels tall or essentially "widescreen aspect ratio in portrait orientation" seems like it would be a bit difficult to use.  I'm also not sure I get your desired end state of "my work laptop showing 2 virtual monitors and the personal PC showing two in PBP".  What are the arrangements/resolutions of these virtual monitors you're trying to achieve in your ideal state?

And even if you're using some application on your systems to carve up the physical display area available to the system into smaller virtual displays, I don't understand what connection that has to the display being in PBP mode.  If you both systems to be shown on the display simultaneously, then you'll be in PBP, regardless of what each source system then does with the "tile" exposed to it by the display.  And if you're in PBP, then you have the KVM function available, at which point you should be able to toggle which system has access to your peripherals.  And if you're NOT in PBP, then you'll only be able to see one system at a time, but that doesn't appear to be what you want.

@david.gray  So you want a single PC to see the U4919DW itself as two or more separate displays? The only way to do that would be to connect the same PC to multiple inputs on the U4919DW and then enable PBP using those two inputs. But since that might not always be feasible and would still only get you two “tiles”, you may want to look into installing Dell Display Manager. It doesn’t let you create totally virtual displays at an OS level, but it does allow you to carve the display up in various ways for application window management purposes.

But the U4919DW has no native ability to split itself into multiple displays within a single input. If you came from multiple physical displays with a KVM and were expecting this single display to emulate that functionality through its built-in KVM, that isn’t how it works. It’s a single physical display and therefore operates as a single display KVM.