I posted the question on here, but received no answer

I recently found out about the Steinmetz Solids, 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 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 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, 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, but for the volume of the cylinders' intersection whose axes are parallel to some solid's diagonal.

I posted the question on here, but received no answer

I recently found out about the Steinmetz Solids, 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 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 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 infinitely many intersecting cylinders with this kind of configuration, will be the unit ball

$ 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, 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, but for the volume of the cylinders' intersection whose axes are parallel to some solid's diagonal.