Local laws for sparse sample covariance matrices without the truncation condition

We consider sparse sample covariance matrices $\frac1{np_n}\mathbf X\mathbf X^*$, where $\mathbf X$ is a sparse matrix of order $n\times m$ with the sparse probability $p_n$. We prove the local Marchenko–Pastur law in some complex domain assuming that $np_n>\log^{\beta}n$, $\beta>0$ and some $(4+\delta)$-moment condition is fulfilled, $\delta>0$.