There is ONLY one way to safely do this: Screws into studs.
I would not trust any kind of "anchors". The problem is that while the basic weight of the furniture itself might be just 44 lbs., by the time you add stuff on the shelves, computer on the desk and you leaning on the desk the weight is much, much more.
The screw holes you show in the picture may or may not really matter. You need to get screws solidly into studs. Measure the thickness of the furniture back, add 1/2" (usually) for drywall and add 1-1/2" into the stud and use screws about that long. Actually, even longer might be better but there is a legitimate concern of wires or pipes going through the middle of the studs.
If the holes match the studs, great. If they don't then you have a couple of choices:
- Drill new holes in the furniture to match the stud locations. That is probably the easiest thing to do.
- Cut wood to match the back of the furniture. Either plywood just smaller than the entire thing, or two strips, one for the top and one for the bottom. Screw the wood into the studs. Screw the furniture into the wood using the existing holes. If you do that then the screws should be thickness of furniture + thickness of added wood plus a little extra.