The problem with HDR illumination is that it is ambient environmental lighting. It is best to provide an infinite light to sharpen and define the directional shadows.  Where EZDome is concerned, the HDRI should provide ambient lighting from the outer dome and a sharp background image of the inner dome. The infinite light (Sun) should be added, as necessary, to simulate the time of day.  This means that the HDRI should be convoluted, or modified, to provide pure, ambient environmental illumination to your scene.  

I have discovered that you can create varying specular and diffuse HDR lighting from the same source image.  But you required third-party software to make the modifications.