geometric computation – GeometricScene extremely slow or cannot complete

I would like to find instances of the following dodecahedral wheel
a geometric figure composed of equilateral triangles and squares

generalized to the situation where the red squares become rhombuses.

I tried this using RandomInstance(GeometricScene), but Mathematica could not complete it the way I was trying. I tried simplifying the Geometric Scence to some smaller subset of constraints (eliminating the outer 6 ring of equilateral triangles):

RandomInstance(
 GeometricScene({a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r,
    s},
  {
   t1 = Triangle({a, b, c}),
   t2 = Triangle({a, c, d}),
   t3 = Triangle({a, d, e}),
   t4 = Triangle({a, e, f}),
   t5 = Triangle({a, f, g}),
   t6 = Triangle({a, g, b}),
   
   s1 = Style(Polygon({b, k, l, c}), Red),
   s2 = Style(Polygon({c, m, n, d}), Red),
   s3 = Style(Polygon({d, o, p, e}), Red),
   s4 = Style(Polygon({e, q, r, f}), Red),
   s5 = Style(Polygon({f, s, h, g}), Red),
   s6 = Style(Polygon({g, i, j, b}), Red),
   
   GeometricAssertion({t1, t2, t3, t4, t5, t6}, "Equilateral", 
    "Clockwise"),
   GeometricAssertion({s1, s2, s3, s4, s5, s6}, "Equilateral", 
    "Clockwise")
   }
  ))

but this also couldn’t complete.

This even smaller subfigure did complete:

RandomInstance(GeometricScene({a, b, c, d, k, l, m, n},
  {
   t1 = Triangle({a, b, c}),
   t2 = Triangle({a, c, d}),
   t9 = Triangle({c, l, m}),
   
   s1 = Style(Polygon({b, k, l, c}), Red),
   s2 = Style(Polygon({c, m, n, d}), Red),
   
   GeometricAssertion({t1, t2, t9}, "Equilateral", "Clockwise"),
   GeometricAssertion({s1, s2}, "Equilateral", "Clockwise")
   }
  ))

enter image description here

but was very slow.

This doesn’t seem to be such a complicated constraints problem, and so I am wondering if there is a better way to do this with GeometricScene?

Thanks for any help you might provide.