I get the same behavior.
You could try a six-column table instead of the three column table with two section-set columns.
You could have no borders at all and put an underscore in the third column in each cell.
I am attaching your document with the latter option for the first three rows. The rest is left as an exercise for the student.
I also modified your headers/footers to use a StyleRef field and to use the Title and Subtitle styles. The StyleRef field gives you the dictionary-style header for the names. I put pagination in a footer.
To use the StyleRef field I formatted your last name cells in a LastName character style.
I also designated your first row as a header row so it would repeat.
Tables
Sections / Headers and Footers in Microsoft Word 2007-2013