Let $Z$ be a fixed $d \times d$ matrix and let $G$ be a random $d \times d$ matrix with each entry i.i.d. $N(0, 1)$.

Is it true that:

$$\mathrm{Tr}(\mathbb{E}_G[ (Z^T + G^T)^\ell (Z + G)^{\ell-k-1} Z (Z+G)^k ] ) \geq 0 \:,$$

where $\ell, k$ are positive integers and $\ell - k \geq 1$?

When $d = 1$, this reduces to:

$$ \mathbb{E}_g[ (z+g)^{2\ell - 1} z ] \geq 0 \:, $$

which is easily verified to be true, since the LHS is of the form $a_2 z^2 + a_4 z^4 + ... + a_{2\ell} z^{2\ell}$ with positive coefficients.