Equivalence between primitive normal bases and field structure of the generalized distributive set

Determine whether the existence of a primitive normal basis of the generalized distributive set D(1,b) over the prime field \mathbb{F}_p is equivalent to D(1,b) being the finite field \mathbb{F}_{p^{l\mu}}, where \mu=\gcd(s,t,n) for the parameters s and t associated with b and 1+b.

Background

The paper studies generalized distributive sets D(a,b) in a finite Dickson nearfield DN_g(q,n), reducing the analysis to sets of the form D(1,b). For nonzero b and 1+b, the parameters s and t determine \mu=\gcd(s,t,n), and the paper proves that \mathbb{F}{p^{l\mu}} is contained in D(1,b). It also establishes that if D(1,b) is closed under field multiplication, then D(1,b)=\mathbb{F}{q^\mu}=\mathbb{F}_{p^{l\mu}}.

The paper defines a primitive normal basis of D(1,b) over \mathbb{F}p as a basis consisting of the p-power conjugates of a primitive element. It notes that the implication from D(1,b)=\mathbb{F}{p^{l\mu}} to the existence of such a basis follows from the primitive normal basis theorem for finite fields. The unresolved direction is whether the existence of a primitive normal basis forces D(1,b) to equal this subfield, rather than merely containing it.