Integrating over a region without singularity gives an error indicating the function has singularity in that region

Question

I have the following 2D region over which I want to integrate a function:

r1  = Sqrt[mη^2 + (Sqrt[w1^2 - mπ^2] + Sqrt[w2^2 - mπ^2])^2] // Expand;
r2  = Sqrt[mη^2 + (Sqrt[w1^2 - mπ^2] - Sqrt[w2^2 - mπ^2])^2] // Expand;
mη  = 0.547;
mηp = 0.958;
mπ  = 0.137;

RegionPlot[mηp - w1 - w2 < Re[r1] && mηp - w1 - w2 > Re[r2], 
   {w1, .1, .25}, {w2, .1, .25}, BoundaryStyle -> Blue, FrameLabel -> {"w1", "w2"}]

Now I want to integrate the following function over this region:

function[w1_, w2_]=Abs[1.05133+ (6.16152 (0.656093- 1.916 w1 - 1.916 w2)
    (0.958- 1. w1 - 1. w2))/(0.921055- 1.916 w1 - 1.916 w2) - 
    (10.1147 (0.656093- 1.916 w1 - 1.916 w2) (0.958- 1. w1 - 1. w2))/
    (1.57895- 1.916 w1 - 1.916 w2) - 55.0594 (-((0.479 w1 (-0.618555 + 1.916 w1))/
    (0.0307393+ 1.916 w1)) - ( 0.479 w2 (-0.618555 + 1.916 w2))/(0.0307393+ 1.916 w2))]^2

When I plot the function it doesn't have any singularity:

Plot3D[function[w1,w2],{w1, 0.137, 0.2445}, {w2, 0.137, 0.2445}, BoxRatios -> {1, 1, 4},
   RegionFunction -> Function[{w1, w2, z}, mηp - w1 - w2 < Re[r1] && mηp - w1 - w2 > Re[r2]]]

But when I use the following command:

NIntegrate[function[w1,w2] Boole[Re[r2]< mηp - w1 - w2 <  
   Re[r1]], {w1, 0.137, 0.2445}, {w2, 0.137, 0.2445}]

Mathematica says it has a singularity. Why?