We thank the referees for their interest in our paper and for their valuable comments that help us to make the paper 1 clearer

We analyzed the multi-layer case beyond what is reported in the submitted paper. Equations to get the optimal error in the multi-layer case are in page 10-11 of the SM. The vertical lines show the PCA and the optimal threshold respectively. Our claims of optimality of AMP are indeed limited to the cases investigated numerically. We will make a statement collecting all the assumptions in the final version.