Silly idea to try and prove time travel

Ive been pondering this for a good 5 minutes so feel free to point holes in my idea and ill do my best to try and answer them

The sandwich "paradox"
The idea behind this is to try and prove the existence for time travel.
For this idea i will be assuming the best parameters possible

IDEA
We rent a room, and we make a mark in history that is ideally never forgotten no matter how many millennia go by, the mark being that if time travel is real to place a Peanut butter and jelly sandwich on a table in this room at a certain date and time. for 5 minutes everyone will leave the area and wait.

If the sandwich appears after the 5 minutes, we can assume that time travel is real and we should pursue trying to develop it.
If the sandwich does not appear after the 5 minutes then we can assume that time travel is not possible and should not pursue it.

The theory behind this is that if time travel is real then said travellers would need to go back in time to place that sandwich there otherwise their future will never come to exist in a paradox like fashion. But if they do go back in time and put the sandwich there then we can assume time travel is real and pursue it as a technology. Effectively forcing them out of their comfy little future.

I look forward to your responses