I have a 2006 Yamaha Roadstar with 104k miles on it. I have never had this problem, but I also ride all year around and commute on my bike. I perform most of the maintenance on my bike. She is just happy with 87 octane regular gas. I also use StarTron about every other tank.
imho, first buy a service manual.
Me personally, I'd drain the tank, carb, fuel line, fuel pump. I have plastic parts in the fuel bowl on my Roadstar so I'd have to pop the carb off and check the float. I'd at least clean the jets with some carb cleaner. Verify the needle moves freely. Verify the condition of the diaphragm and manifold. Don't remove the manifold, just inspect with a flashlight and mirror. Looking for cracks.
Put the battery on a Battery Tender until the light turns green.
Add StarTron to your tank then the recommended fuel for your bike. (Startron first, fuel second - this mixes everything).
Put bike in neutral, open the petcock, open enricher circuit, key on, verify lights are on, master cutoff switch to on, listen for fuel pump to emit a series of clicks. Wait for clicks to stop. Depress started button for no more than 5 seconds. If bike doesn't start try again. If after 3 tries the bike doesn't start - shut everything off. Verify fuel, air, spark.
You may (should) give a Suzuki mechanic a call for their opinion. Having a good relationship with a decent shop is like have a good relationship with your local pro shop. Both are invaluable.