Local limit theorem and recurrence for a random walk with unbounded jumps

Let $\{S_n\mid n\geq 0\}$ be a random walk with unbounded jumps, and write $S_n=\sum_{k=1}^nX_k$. For $i=1,2$, suppose that

$$\begin{cases} \Pr(X_{n+1}=\pm e_i\mid S_n=(0,0))=1/4,\\ \Pr(X_{n+1}=\pm n e_i\mid S_n\neq(0,0))=\operatorname{const.}\,|n|^{-3}. \end{cases}$$

Define $V_n(t)=(n\log n)^{-1/2}S_{\lfloor nt\rfloor}$ for $t\in[0,1]$. The preceding theorem gives $V_n(t)\Rightarrow C W(t)$ weakly in $C[0,1]$, where $W$ is standard planar Wiener process and $C>0$.

Unbounded-jump random-walk conjecture. In this setup, the local version of the limit law for $V_n(1)$ is also true, and the random walk defined above is recurrent.

These assertions would supplement the stated global limit theorem for the non-standard $\sqrt{n\log n}$ domain of attraction. The supplied text gives no resolution status for either assertion.