June's Towel of the Month, like May's, is also Atwater-Bronson lace (aka Lace Bronson), but instead of one of the blocks being turned, this draft uses two "normal" blocks of lace that are woven to make squares in an all-over lace pattern over the towel.
The threading goes like this: 4 ends of plain weave for the selvedge or border, then start the repeating sequence: 10 ends plain weave, 12 ends (2 units) of lace block A, 12 ends lace block B, 12 ends lace block A. So the repeat has 46 ends. Repeat those 46 ends 12 times (552 ends). Then finish with 10 ends of plain weave to balance, and 4 ends of selvedge/border. Total 4 + 552 + 10 + 4 = 570 ends.
The warp is bleached (if you can find it - I used1/2-bleached) 40/2 linen, and the weft is bleached 20/1 linen.
The treadling repeat in the Barrett article says to put 30 picks of plain weave between your lace block squares, but you might want to try 10 to make it the same "as drawn in".