A different way of looking at the focus breathing of the 70-200mm f/2.8 VRII

The OP is broadly right

When a lens changes effective focal length in close focus the physical size of the aperture does not change.

The effective close focus aperture changes in the same way as the effective focal length.

In close focus the Mk II is about one stop effective faster than the original version.

Whether the original version was effective f2.8 at minimum focus is another topic

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Leonard Shepherd
Many problems turn out to be a lack of intimate knowledge of complex modern camera equipment.