Average shifted convolution sum for $GL(d_1)\times GL(d_2)$

We study the average shifted convolution sum $$B(H,N):=\frac{1}{H}\sum _{h\sim H}\sum _{n\sim N}{A}_{{\pi }_1}(n)A_{{\pi }_2}(n+h),$$ where $A_{{\pi }_i}(n)$ denotes the Fourier coefficients of a Hecke–Maass cusp form ${\pi }_i$ for $\mathrm{S}\mathrm{L}(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. We establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\epsilon }$ for any $\epsilon >0$. For the cases $d_1=d_2+1$ and $d_1=d_2$, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when $d_1=d_2$, we reach the critical threshold $H\ge N^{1-2\mathrm{/}d+\epsilon }$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect.