Tapered particle system

I am trying to model the fishing fly below:


For the “body” of the fly I have created a particle system using an object:



I would now like to “cut” the “hairs” in a cone shape tapering from the eye of the fly to the tail. I have tried to convert the particle system in the modifiers tab but then I get 2500 new hair objects! Also when I try to Ctrl J these hair objects a whole new stack of “hair” appears out of nowhere.
What will be the easiest way to taper the hair. Can it be done in the particle system or possibly by using weight paint o something?

Further to the above I have found this thread and thought my problem was solved: https://blenderartists.org/forum/showthread.php?381926-Particle-Edit-Shape-Object-Shape-Cut
However, I just cannot seem to get the correct result. I get the hair to be cut but afterwards the hair is thinner! and although all is shown to be cut to the shape of the object when in Particle edit mode, as soon as I switch to object or render mode some hairs re-appear!
Please help!DDD part.blend (1.13 MB)