Consider a sequence of integer valued indentically distributed centered independent random variables $X_1, X_2, \ldots$ with the additional condition that the support of $X_1$ is aperiodic. Suppose that the distribution of $X_1$ is in the domain of attraction of a stable law of index $\alpha \in (1,2]$. This means that there exists a sequence $(B_n)_n$ such that $(X_1+X_2+\cdots+X_n)/B_n$ converges in distribution to a stable law of index $\alpha$.
Then the following convergence, known as the Local Limit Theorem (see e.g. I.A.Ibragimov, Yu.V.Linnik, Independent and stationary sequences of random variables, Theorem 4.2.1), holds: $$\lim_{n \rightarrow \infty} \sup_{k} \left | B_n \mathbb{P}[X_1+\cdots+X_n=k]-g(\frac{k}{B_n})\right| =0,$$$$\lim_{n \rightarrow \infty} \sup_{k} \left | B_n \mathbb{P}[X_1+\cdots+X_n=k]-g\left(\frac{k}{B_n}\right)\right| =0,$$ where $g(x)$ is the density of some $\alpha$-stable distribution.
When $X_1$ has a finite second moment (which implies $\alpha=2$), a refinement of this theorem is known, which states that: $$\lim_{n \rightarrow \infty} \sup_{k} (1 \vee \frac{k^2}{n}) \left| \sqrt{n} \mathbb{P}[X_1+\cdots+X_n=k]-p_{\sigma^2}(\frac{k}{\sqrt{n}})\right|=0,$$$$\lim_{n \rightarrow \infty} \sup_{k} \left(1 \vee \frac{k^2}{n}\right) \left| \sqrt{n} \mathbb{P}[X_1+\cdots+X_n=k]-p_{\sigma^2}\left(\frac{k}{\sqrt{n}}\right)\right|=0,$$ where $\sigma^2$ is the variance of $X_1$ and $p_t(y)=\exp(-\frac{y^2}{2t})/\sqrt{2 \pi t}$$p_t(y)=\exp\left(-\frac{y^2}{2t}\right)/\sqrt{2 \pi t}$.
Is there an analog of this strong local limit theorem in the case $1 < \alpha < 2$? For instance, is it true that: $$\lim_{n \rightarrow \infty} \sup_{k} (1 \vee \frac{k^\alpha}{B_n^\alpha}) \left | B_n \mathbb{P}[X_1+\cdots+X_n=k]-g(\frac{k}{B_n})\right| =0 ?$$$$\lim_{n \rightarrow \infty} \sup_{k} \left(1 \vee \frac{k^\alpha}{B_n^\alpha}\right) \left | B_n \mathbb{P}[X_1+\cdots+X_n=k]-g\left(\frac{k}{B_n}\right)\right| =0 ?$$
