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How can we create a 3D fractal flower in Mathematica?

Posted 6 months ago
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In a 2D space, we can use simple affine equations to create fractal shapes that resemble Barnsley's Fern. Now, there is something interesting about these shapes that is discussed in a MathOverflow post. It says:

If you allow for a larger class of functions (stochastic, 3-dimentional real maps, and introduce a log-density plot and color each point according to orbit history, the possibilities are endless (image created by Silvia Cordedda):
Fractal Flower

Now the problem is, recreating such images needs much more details that I am not aware of. Obviously, one cannot simply extend the generating equations of Barnsley's Fern to 3D and hope that something magical like this pops out. So I need some guidance from the experts on this site about this. Have you ever done something like this in Mathematica?

(I have also asked this question on Mathematica SE which didn't attract any answers. So I decided to give it another try in here. Thanks in advance).

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