Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k \times k$ matrix $C(W)$ by setting $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$.

Question. Is there a high-probability good lower-bound for the smallest eigenvalue of $C(W)$ ?

Related: https://math.stackexchange.com/q/4005530/168758

N.B. I'm familiar with standard RMT.