The antenna is a Jpole and has no relationship to the coaxial sleeve antenna. The bottom of the sleeve on a coaxial antenna is high impedance, the top is a current maxima- just the opposite of the Sigma 4. In addition the sleeve on the coaxial antenna has, because of skin effect opposite currents on the inside and outside.
The reason Herb flared the 1/4 wave arms on the Sigma IV was threefold fold:
1. It made it mechanically easier to build the antenna
2. It likely acts as a transmission line whose characteristic impedance increases as the separation increases. This would be more important if this were a more traditional transmission line with dielectric, as it would reduce the losses in this matching section.
3.It looked cooler
It could still benefit from a means to choke off common mode currents as there is no reason, the support mast and feedline would not have current on them. A quick EZNEC model will show this.
Model the antenna and then put a quarter wave wire from the bottom of the antenna extending downward (admittedly, a worst case scenario, but done to prove a point). That wire would represent the mast and/or the outside of the coaxial shield. Rerun the model and look at the currents on that quarter wave wire.
I realize I really didn't address much of what Dale said directly so I'd like to do that here.
1) How is it mechanically easier to construct a much longer, 30 foot free standing radiator that eliminated one insulator to add several more that now support the cone?
2) If the purpose is to match the impedance why was there no success in arriving at a 50 ohm match and why did Herb decide to use a gamma to achieve the match? It's more likely that the design of this cone was primarily to confine the out of phase radiation on the vertical inside of it with the added benefit of a second current maxima point that is in a constructive phase.
3) I'll give you the fact that "It looked cooler" but this is not what Herb describes in his patent where he claims 1 db over a 5/8 wave groundplane.
The idea that a choke at the feedpoint would help or that adding a 1/4 wave counterpoise would show a worst case scenario are contradicted by the CST model and field tests. If you understand the CST representation of the current phase being radiated from the outside of the cone is one that constructively adds to the phase of the 1/2 wave above it, why would you think extending a counterpoise in any other direction would change this phase? Phase is determined by the electrical length from the source.