For quite some time I have had a random problem where the editor destroys nested tables. It has always seemed to happen when adding or removing paragraph tags/carriage returns. Well finally I figured out a consistent way to produce the problem and I have repeated it in your online demo.
My sample HTML is found below. To repeat the problem, open the HTML in the editor. Now place the cursor after RESEARCH but note there is a space after the word Research. You need to be to the right of that space. In reality there is no space there but I guess the span tags make it appear as though there is a space?
Now, hold the shift key and hit the left arrow so the "space" to the right of RESEARCH becomes highlighted.
Now hit enter or delete.
Now save and the table structure is destroyed. The table on the right side is moved to the bottom.
Please help!