Check for something that might have fallen down into the fuel tank, and which slides over the tank outlet, shutting off the flow of fuel! BTDT!
Also, check to see if the breather hole in the gas cap might be plugged by a bug egg! BTST!
could be the ignition coil losing its serviceability. They do tend to go bad occasionally, at high miles/ hours of use.

On these, cover the basics. Ensure no water or debris in the carb bowl, try a new spark plug, ensure that the solenoid below the bowl activates when the ignition is on. If all is good, either the ignition coil is failing or the fuel pump is quitting. This engine can do either so you will need to determine if it starts to starve for fuel or not.

Is it always on the slope, or just the first time?

I don't know if these have low oil sensors, but if so maybe your oil is low???

Good luck