For what it's worth... typically I get extra paper around the artwork so that I can wrap it around the board's edges.
I usually put the art in position on a board, and use it as a guide for cutting the holes. I hammer a punch to force my drill bits to be perfectly aligned.
After making the cuts in the board I spray a healthy coat of archival quality spray adhesive on the back of the art. Set it on the board and use a bondo scraper or other hard plastic to push out any bubbles. Roll the bottom edge under the front of the board, stretching lightly and staple it. I cut triangles in the corners, stretch the sides and roll under, stapling to the board. I do the same thing with the back of the art. I make sure the staples are in tight by tapping them with a hammer. Between the staples and glue the art stays down for good.
Looks like your art is flush, so you can't use my method, but this is how I do it, and I've never had issues. I usually cut the holes out of the art after the glue dries with a new exacto blade using the wood edges as a guide. Sometimes before I cut I flip the board and brush a light coat of glue right along the edges of the holes next to the paper to make sure that when I cut the paper it really stays tightly glued after it's cut.
Also... I use Game on grafix that advertises on this site. After 10 years I finally stripped the art off a panel (was still in great shape) so that I could pull out a flush mounted trackball. It was still stuck on solid.