Does the use of a gunner major action always preclude the use of the Snap Shot minor crew action?

In Starfinder, during starship combat, the Snap Shot minor crew action allows a crew member who has taken a major crew action earlier in the round to fire a starship weapon during the gunnery phase. The rules state:

You can fire one of your starship’s weapons with a –2 penalty to the gunnery check. You can take this action only if no other gunner actions have been taken during the gunnery phase (including snap shot).

The wording of the second sentence raises questions. Does it disallow Snap Shot actions completely if another character intends to take a gunner major action? Does it disallow Snap Shot actions unless they’re taken before all gunner major actions for the round? Or does it simply disallow Snap Shot if the character attempting it has already carried out a gunner major action?

The scenario I am facing in play is this: a pilot has completed a pilot major action in the Helm phase and positioned the ship to put the enemy ship in the port quadrant. There are two gunners on board. The first gunner has fired at the enemy ship with a turreted weapon. The second gunner has fired at the enemy ship with a port arc weapon. The pilot’s player points out that the ship also has a forward arc weapon that has the broad arc property–meaning that it can target ships in the port or starboard arcs with a -2 penalty. He wishes to Snap Shoot the broad arc weapon at the ship in the port arc with a cumulative penalty of -4 (-2 for firing outside the weapon’s normal arc and -2 for the fact that it is a Snap Shot). This seems like a reasonable request to me, but do the rules preclude it? If they do, then had the pilot declared this intention before the other gunners rolled, would he have been able to do it within the rules?