I did this question just now and wonder if my interpretation is correct or just complete nonsense (though it seems to tie in well with the previous bit of the question):
The first part of the question talked about a 1D diatomic chain, so when we consider longitudinal modes at the k=(1,0,0) boundary in 3D, we can think of this as like a 1D chain of atoms, as if we were just considering a line of atoms along the x-axis of our crystal. In the Germanium fcc structure they describe, along the x-axis of the crystal the atoms are all equally spaced, so we could think of this as being just a monoatomic chain, for which there is no splitting, so the states are degenerate.
If we look at a 1D chain of atoms that points along the (1,1,1) direction of the lattice, the spacing between atoms is not uniform, which we can model with alternating spring constants. This case is like the diatomic model studied in the first part of the question, for which there is splitting at the boundary, so there is no degeneracy at the BZ boundary.
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