If Lobot is in an engagement with an undamaged Executor, does Executor's power "cannot be targeted by enemy card effects" also protect against Lobot's power to treat all combat icons as edge-enabled?

In other words, does Executor keep its 3 edge-independent unit-damage even if DS loses the Edge battle?

In other words, does Executor keep its 3 edge-independent unit-damage even if DS loses the Edge battle?

No, it doesn't. The Executor can't be targeted by enemy card effects (Rebel Assault), cards that don't make you specify a target can still affect the Executor (Lobot, X-Wing Escort).

Lobot doesn't Target. So yes it works fine.

Thanks for the clarification.

That "target" specifier is an interesting detail. And it makes Lobot even more interesting to play.