If you are thinking clearly you will realize you don't need a solid to begin with.
The reason is Split is the wrong command, Trim is the correct command to use.
If you think about it the answer to your question is sort of obvious, After you split one of the objects its no longer the same object. If you want to do what you suggest then make a copy of the first object you split to use to split the second.
But using Split command here is bad technique. Your model is an example of why its bad to use Split and good to use Trim command. Either one might succeed for doing the cutting, but if you want the objects to join after the cutting the Trim command is more reliable to make sure that happens correctly. The short explanation of why, is the Trim command calculates both the new boundaries at the same time and that pretty much guarantees the 2 polysurfaces will have matching boundaries that join up. If you do 2 separate split commands there is no guarantee that the edges will match up and join, particularly at the end points of the edges (vertices).
Doing 2 split operations on intersecting polysurfaces is a good way to end up with failed joins or the tiny orphaned edge loops at the end of edges that are supposed to match.