If drilling holes in the wall is a permanent or insurmountable objection, here is what I would do after thinking about this for a bit:
You need masking tape, superglue (cyanoacrylate or CA glue), and some small strips of thin wood. You'll be using the superglue-and-tape trick.
Apply tape to the wall at the height of the tabletop where the strips of wood will go. Two or three pieces of wood, spaced along the long edge of the table, should be sufficient. Each one only needs to be a few inches long (about 10cm) and the wood can be quite thin1.
Apply masking tape to both sides of the strips. Apply masking tape to the relevant positions on the tabletop's rear edge. Burnish the tape down well.
Add strings of glue to the tape on the wall and glue the strips into position. After a minute or two do the same on the front of the wood strips, then push the table backwards into them and press hard for a few moments.
And that's it, you're done.
Your table should be completely stable after this without any holes having to be drilled and without having to futz around trying to figure out and then implement any bracing options that legs of this type make difficult.
In addition to being easily reversible, this is also an inexpensive solution2 especially if you use dollar-store superglue, which is perfectly sufficient for this purpose. Ditto the masking tape. If you're in the US or Canada I understand that stirring sticks can be free in many places3. So total cost here is perhaps five bucks :-)
1 If it would allow enough space for your cabling stirring sticks from the hardware store/big box would do fine.
2 I was initially going to recommend using Command Strips to attach the wood to the wall until I checked their price o_O
3 If not raid a dumpster outside a construction site or home reno for a few pieces of discarded furring strips; I'm not even kidding.