Yeah, they will engrave. You’ve got a bunch of heart shapes with a Black variable width Stroke and White Fill.
You know the simplest thing to do is just rasterize it. Gives great results if you use a high PPI.
Once you have it looking the way you want it, select all the flowers (not the Red Cut Line) and click on Object > Rasterize > High Resolution and Transparent Background. That should do it.