Can Jax's Quick-Draw be played against Falling Star to increase Might on the chain?

Yes, you can play an Equipment card with Quick-Draw (granted by Jax, Unmatched) in response to Falling Star to increase a unit's Might.

Here is the breakdown of how this interaction works:

The Sequence

1. Falling Star is played and placed on the chain.
2. Because Falling Star is on the chain, the game state is "Closed."
3. Jax, Unmatched grants your Equipment Quick-Draw. According to Rule 745.1.b and 745.1.c, Quick-Draw allows you to play Equipment from your hand using [Reaction] timing.
4. You play an Equipment card as a [Reaction] to Falling Star.
5. The Equipment enters play immediately (it does not use the chain to enter).
6. The Quick-Draw triggered ability ("When you play it, attach it to a unit you control") is placed on the chain as a new link above Falling Star.
7. If the Equipment provides a Might bonus (e.g., via a Might Bonus in its lower right corner, per Rule 136), that bonus is applied to the unit as soon as the Equipment attaches.

Important Nuances

• Timing: You are playing the Equipment as a [Reaction] to the Falling Star spell. Because the Equipment attaches as part of the Quick-Draw trigger, the Might bonus will be active before Falling Star resolves.
• Resolution: The chain resolves in Last-In, First-Out (LIFO) order. The Quick-Draw trigger will resolve first, attaching the gear and granting the Might bonus, followed by the resolution of Falling Star.
• Might Bonus: If the Equipment provides a static Might bonus, it will increase the unit's Might immediately upon attachment. If the unit's Might is increased to a level that survives the 3 damage from Falling Star, the unit will remain on the board.

Summary: You can use the [Reaction] timing granted by Quick-Draw to attach gear and boost a unit's Might while Falling Star is waiting on the chain.