The l-Hensel Lifting Assumption

In one of my previous posts, i talked about the possibilities to insert backdoors in prime numbers (beside the property that $p-1$ is smooth). One of the potential ideas for a backdoor was to generate primes that allow an easy lift from $\mathbb{Z}/p$ to $\mathbb{Z}/{p^2}\mathbb{Z}$. But under the consideration of the $l$-Hensel Discrete Logarithm Assumption, this should be actually hard in the general case.

Because if you could successfully lift the element $e\equiv g^x\pmod{p}$ to $E\equiv g^x\pmod{p^2}$, then you could, as long as $x < \mathsf{ord}_p(g) = r$, compute
\begin{align*}
\left(g^x\right)^{r} \equiv (g^{r})^x \equiv (1+p\lambda)^x \pmod{p^2}
\end{align*}
Next, the Binomial Theorem gives
\begin{align*}
(1+p\lambda)^x \equiv \sum^x_{i=0}\binom{x}{i}p^i\lambda^i \equiv 1 + xp\lambda\pmod{p^2}
\end{align*}
and hence
\begin{align*}
\frac{E^{r} - 1}{p\lambda} \equiv x \pmod{p}
\end{align*}