Yeah, that would probably do it...
As for abatement... do what you need to do for milivolts anyway. Probes suck, use direct coax connections wherever possible. Build deadbug/manhattan on copper clad instead of solderless breadboard, use lots of BNC connectors.
That'll at least reduce things to internal noise, which that switching noise could very well be. A shame, oh well. If you're desperate, you could try adding more filtering inside the scope -- ferrite beads and ceramic caps along the power supply wires to the main board, say, and shielding around the PSU if it doesn't have some already. (The power supply in my TDS460 is literally more than half inductors, it's ridiculous... but I bet it's at least as quiet as needed, too!)
Tim