I posted the question on [here][1], but received no answer

I recently found out about the [Steinmetz Solids][2], obtained as the intersection of two or three cylinders of equal radius at right angles. If we set the radius $ = 1 $ the volumes are respectively

$ V_2 = \frac{4}{3}\cdot4 $

$ V_3 = \frac{4}{3}\cdot6(2-\sqrt{2}) $

A natural question is to ask for the volume of the shape created by the intersection of more cylinders whose axes intersect all at a single point: Moreton Moore wrote an [article][3] where he calculates the volume of the intersection of $ 4 $ and $ 6 $ cylinders with the axes passing through the center of the opposite faces of the octahedron and dodecahedron respectively

$ V_4=\frac{4}{3}\cdot9(2\sqrt{2}-\sqrt{6}) $

$ V_6=\frac{4}{3}\cdot4(3+2\sqrt{3}-4\sqrt{2}) $

[In another post][4] was given the answer for the $ 10 $ cylinders case:

$ V_{10} = \frac{4}{3}\cdot\frac{15}{4}(24 + 24 \sqrt{2} + \sqrt{3} - 4\sqrt{6} - 7\sqrt{15} - 4\sqrt{30}) $

It's immediate to note that the limiting volume of infinite intersecting  cylinders will be the unit sphere

$ V_\infty = \frac{4}{3}\cdot\pi $

So my question is if a formula is known for the general volume $ V_n $. The question was already asked [here][5], but no response was given, so after $10$ years I would like to know if anything new was discovered about that. Writing $V_n = \frac{4}{3}\cdot\sum_{i=1}^N a_i $  how can I calculate the algebraic coefficients $ a_i$ without calculating the integrals for each case? The corresponding serie would be $ \sum_{i=1}^\infty a_i = \pi$

I'm not asking for the minimal volume among all the possible configurations of $n$ cylinders, because it would be rather obvious how to calculate the volume in [that case][6], but for the volume of the cylinders' intersection whose axes are parallel to some solid's diagonal.


  [1]: https://math.stackexchange.com/questions/4861254/the-intersection-of-n-cylinders-in-3-dimensional-space
  [2]: http://en.wikipedia.org/wiki/Steinmetz_solid
  [3]: http://www.jstor.org/discover/10.2307/3615957?uid=2134&uid=2&uid=70&uid=4&sid=21103329679973
  [4]: https://math.stackexchange.com/questions/1737635/volume-of-the-intersection-of-ten-cylinders
  [5]: https://mathoverflow.net/questions/154386/the-intersection-of-n-cylinders-in-3-dimensional-space
  [6]: https://mathoverflow.net/questions/430504/intersecting-cylinders-around-a-sphere