φ-triviality of extended structures on Taft algebras T_n(ω)

Prove that for the Taft algebra T_n(ω)=k⟨g,x⟩/(g^n−1, x^n, gx−ωxg) with primitive n-th root of unity ω, every extended Frobenius structure (φ,θ) has φ=id and θ lying in the k-linear span of {g^j x | 0≤j≤n−1}.

Background

The paper proves for the special case T_2(−1) that all extended Frobenius structures are φ-trivial with θ in kx⊕kgx, and provides explicit Frobenius/extended data. This mirrors the observed behavior for other small examples and motivates a general pattern.

The conjecture asserts that the φ-trivial phenomenon persists for all Taft algebras T_n(ω), with θ constrained to the span of skew-primitive elements g^j x, thereby extending the n=2 classification to arbitrary n.