Chainmail is often structured in bugs, which is the smallest unit of 4-in-1 weave that looks like chainmail. One bug's width of chainmail is 3 rows wide - two bugs is 7 rows. In fact, a bug is actually 4 rows, not 3, except that one of the rows is what you create when you connect two bugs together. A half a bug, therefore, is 2 rings wide. Box chain can be created by taking a one-bug-wide strip and connecting it to itself.
I now see that the same concepts for European mail apply to Persian mail as well. 3-in-1 "half persian" is, in fact, half a Persian bug wide, or 2 rings. In extending this to 3 rings wide, I found it trying to contort itself into a position to knit it to itself, which would clearly produce Persian box chain in this case.
Unfortunately, 5/16" 16gu rings are too small to make Persian fabric, so to speak. I shall attempt that some other time.