If the trees have strong highlights under HDRI sunlight then they would have exactly the same highlights in real world. It is not a bug, the problem is that people tend to make materials a LOT more reflective than they are in real world, so there's your problem ;)
Matching real world reflection and glossiness of an average leaf material will do the trick for you. If you for any reason need leaves to look like glass, such as in your case, then you can alternative duplicate your bitmap, open the HDR/EXR map with only 16 or 8 bit depth... and then put the bitmap opened at lower bit depth into reflection environment override slot.
The difference with reality is the pixel scale. I will try to render it at a much bigger res to verify that.