Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?

$$E[\|C_n\|_F^2]=d^{n+1}$$

Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof?

$$E[\|C_n\|_F^2]=d^{n+1}$$

This follows from discussion on math.SE on the moment method, but unclear how to adapt it to this, since the moment method requires fixing $n$.