3D class 4 [cellular automaton] rules

With a cubic lattice of the type shown on page 183, and with updating rules of the form

LifeStep3D[{p_, q_, r_}, a_List] := MapThread[If[(#1 == 1 && p <= #2 <= q) || #2 == r, 1, 0]&, {a, Sum[RotateLeft[a, {i,j,k}], {i,-1,1}, {j,-1,1}, {k,-1,1}] - a}, 3]

Carter Bays discovered between 1986 and 1990 the three examples {5, 7, 6}, {4, 5, 5}, and {5, 6, 5}. The pictures below show successive steps in the evolution of a moving structure in the second of these rules.