The triviality conjecture for extensions of Taft algebras

Let $n$ be a positive integer, let $\omega$ be the parameter in the Taft algebra, and define

$$T_n(\omega):=\Bbbk\langle g,x\rangle/(g^n-1,x^n,gx-\omega xg).$$

An extension consists of a Frobenius automorphism $\phi$ and an element $\theta$; call it $\phi$-trivial when $\phi=\operatorname{id}_{T_n(\omega)}$. The Taft-algebra conjecture. Every extension of $T_n(\omega)$ is $\phi$-trivial, and its additional element satisfies

$$\theta\in \Bbbk x\oplus\Bbbk gx\oplus\cdots\oplus\Bbbk g^{n-1}x.$$

The paper proves the assertion for $n=2$ and proposes the displayed description for general Taft algebras; the general case remains open in the supplied text.