No, not much... In fact the other way is more likely...
The price went out of sight as fewer folks shot the .410. The highest I remember (no ammo hoarding times) was High 12 bucks a box for remington "Express" or Winchester "Super X" both not a "premium ammo" by any stretch.
Both were around #6 shot. The same in 12 gauge the same day were around $5... maybe closer to $6.
If they sell a brazillian of those pistols then maybe the demand will force more supply and then it will level off and prices may possibly get back down when the shooters realize the MASSIVE profits the companies get compared to the far heavier larger bore shells.