While sitting in a doctor's office, I just formulated this hybrid
derivation - A lot of Gauss' law with superposition added to taste:
1) s1 + s2 = -s3 -s4 From the problem statement.
2) s2 = -s3 (Both Gaussian caps in conductors)
3) s1 = -s4 From 1) and 2)
4) E = s2/epsilon the field in the central region. (one cap in conductor,
other cap in central region)
5} GL has told us that the field due to any single charge layer is
E=sigma/(2epsilon).
6) The field given in 4) is completely accounted for by the layers s2 and
s3.
7) Therefore the contributions of s1 and s4 to E must cancel in the
central region.
8) This requires s1 = s4
9) 3) and 8) force s1=s4=0.