Tammes cylinders

Arrange $k$ infinite cylinders of unit radius so that they are all tangent to a common sphere and no two overlap. How small can the sphere be? For $k=6$ there are a couple of natural arrangements around a sphere of unit radius. The first one is with all cylinders parallel and tangent at the equator:

The other one has cylinders tangent at the six vertices of an inscribed regular octahedron:

Note that the first configuration is not stuck in place. Any three adjacent cylinders can start to rotate together:

Even though the configuration is not rigid, there does not seem to be a way to decrease the radius of the central sphere while keeping all the tangencies. Is a unit-radius sphere the smallest one that allows six tangent unit cylinders without overlaps?

Well, apparently not. The following configuration, found by Moritz Firsching, has six unit cylinders around a sphere of radius $0.952690$. The configuration does not seem to have any symmetry. Is there a way to arrange six unit cylinders around an even smaller sphere?

Further reading: MathOverflow.net "How many unit cylinders can touch a unit ball?"

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