Spectral norm bound for the product of random Fourier-Walsh matrices

We consider matrix products of the form $A_1(A_2{A}_2{)}^{\mathrm{\top }}\dots (A_m{A}_m^{\mathrm{\top }})A_{m+1}$, where $A_i$ are normalized random Fourier-Walsh matrices. We identify an interesting polynomial scaling regime when the operator norm of the expected matrix product tends to zero as the dimension tends to infinity.