No. Since
Eternal Scourge is an exiled card and therefore has no controller, 109.5 says that the word “you” on it refers to its owner. Therefore, the only player who has permission to cast it is its owner.
The clause of 109.5 you're referencing applies only to an object a player is attempting to play/cast/activate, and as described in 601.2, in order to begin the process of casting a spell, you must be legally allowed to cast it. The opponent does not have permission to begin to cast Eternal Scourge, so they cannot attempt to cast it.
The relevant rules:
109.5. The words “you” and “your” on an object refer to the object’s controller, its would-be controller (if a player is attempting to play, cast, or activate it), or its owner (if it has no controller). For a static ability, this is the current controller of the object it’s on. For an activated ability, this is the player who activated the ability. For a triggered ability, this is the controller of the object when the ability triggered, unless it’s a delayed triggered ability. To determine the controller of a delayed triggered ability, see rules 603.7d–f.
601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. Casting a spell includes proposal of the spell (rules 601.2a–d) and determination and payment of costs (rules 601.2f–h). To cast a spell, a player follows the steps listed below, in order. A player must be legally allowed to cast the spell to begin this process (see rule 601.3), ignoring any effect that would prohibit that spell from being cast based on information determined during that spell’s proposal. (Such effects are considered during the check detailed in rule 601.2e.) If, at any point during the casting of a spell, a player is unable to comply with any of the steps listed below, the casting of the spell is illegal; the game returns to the moment before the casting of that spell was proposed (see rule 720, “Handling Illegal Actions”).