image processing - Help finding the centre of a faint, incomplete circle?

so I found a program in the answers to this question that I thought was brilliant, so here it is applied to my situation: I have the image and here's the code:

img=https://i.stack.imgur.com/MRlBH.png;
img = GaussianFilter[i, 2]

squaredError = 1/2 ({cx - x, cy - y}.{-gy, gx})^2;


errDerivative = Expand[D[squaredError, {{cx, cy}}]];
linearSystem = {{D[errDerivative, cx],


D[errDerivative, cy]}, -errDerivative /. {cx -> 0, cy -> 0}}

gradientX = ImageData@GaussianFilter[img, 1, {0, 1}];
gradientY = ImageData@GaussianFilter[img, 1, {1, 0}];
xArr = Array[N[#2] &, Dimensions[gradientX]];
yArr = Array[N[#1] &, Dimensions[gradientX]];

ls = Total[
linearSystem /. {gx -> gradientX, gy -> gradientY, x -> xArr, y -> `yArr},{-2, -1}];`
center = LinearSolve @@ ls;


center[[1]] -= 1;
center[[2]] = Length[gradientX] - center[[2]];

maxRadius = 400;
polar = ImageTransformation[img, 

  center + {Cos[#[[1]]], Sin[#[[1]]]}*#[[2]] &, {360, maxRadius}, 
  DataRange -> Full, 
  PlotRange -> {{0 \[Degree], 360 \[Degree]}, {1, maxRadius}}]

radiusStrength = Mean /@ ImageData[polar, DataReversed -> True];
peakX = Position[
    MapThread[#1 > #2 && #1 > #3 &, {radiusStrength, 
      RotateLeft[radiusStrength], RotateRight[radiusStrength]}], 
    True][[All, 1]];

peaks = SortBy[Transpose[{peakX, radiusStrength[[peakX]]}], Last];
ListLinePlot[radiusStrength, PlotRange -> .8, 
 Epilog -> {Red, Point[peaks[[-6 ;;]]]}]

Show[img, 
 Graphics[{Red, Dashed, Circle[center, #] & /@ peaks[[-8 ;;, 1]]}]]

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Fantastic, exactly what I want this code to do for this image. If I crop the image so I only have the top half of the circles it still works great. However, when I try out this image, it shits the bed: This is the final output

Do you guys have any suggestions for a more robust algorithm to find the centre?