# Excluding region in ContourPlot

I want to create a contour plot of a function that goes to infinity at x=1 and z=0 as can be seen by plotting it for x=1:

zmax = 4;
fun[x_, y_, z_, xp_, yp_] := -(3*z^3)/(2*Pi*(z^2 + (x - xp)^2 + (y - yp)^2)^(5/2));
Plot[fun[1, 1, z, 1, 1], {z, 0, zmax}]


The following code

ContourPlot[fun[x, 1, z, 1, 1], {x, -2, 3}, {z, 0, zmax},
AspectRatio -> 1/2, PlotPoints -> 50, Frame -> True, ColorFunction -> GrayLevel,
PlotLegends -> Placed[BarLegend[Automatic, 10, LegendMarkerSize -> 350], Below],
PlotRange -> {Automatic, Automatic, {-0.5, 0}}]


produces a contour plot with a white area around the singularity:

The question is: how to get rid of the white area. I would like to exclude the region around point {1,0} from the calculations and set its colour to black (or any other). I tried adding something like

Exclusions -> {{x < 1.5, x > 0.5, z < 0.5}}, ExclusionsStyle -> Black


but it didn't change anything.

When I see this right, then you don't need this. The inner white "circle" is already clipped. Therefore, the only thing you have to add is ClippingStyle -> LightGreen and you get