Hi, I just gonna give some thoughts.
The region is a wedge below in the first quadrant.
The function integrates the function along the line .
It seems to me that the integrand could take negative values as long as it overshoots fast enough close to the region since you always have to integrate that part.
I think you can "build" such a function slice by slice ( by ). You could take some function that starts at zero, assume negative values until where it goes back to zero. Then it assumes large positive values from to . Find some way to make that -dependent, and you should have a counter-example. You could take for example a which period (and amplitude) varies with for the lower part and a sharp for the upper part.