I'm certain there isn't a requirement. Whether there is a preference is another issue... You could well imagine that the stabilization by hyperconjugation is slightly different in the two transition states leading to the common intermediate, as well as the two starting materials having different energies, so it's likely the rates would differ to some extent.
Just thinking out loud, there are two main factors I can think of that would affect the rate. Assuming the t-Bu remains as an equatorial 'lock' throughout:
1. If the halogen is axial it will experience greater steric destabilization, raising the energy of the SM and making loss of the halogen more favourable to relieve strain.
2. If the halogen is equatorial, then adjacent electron-rich C-C bonds can donate into the C-X sigma*, weakening the bond and making loss of the halogen easier.
I don't know which effect wins, but I'd be happy to hear people's thoughts?