Scene Economy: When an Interstitial becomes something else, who owns it?

There’s mention about scenes developing organically, for instance an interstitial scene becoming a building or conflict scene. If the player who drives the scene from interstitial to conflict is not the player who started the interstitial scene (for instance a conversation is escalated to a DoW or Firefight), the instigator obviously gets the conflict scene charged against their budget but does the interstitial scene still count against the first player?

My gut reaction would be that if a scene is escalated to a conflict that it loses any prior status as a non-mechanics scene (interstitial or color) and as such is not charged against the interstitial or color scene budget.