Bounding the inner product of a vector of correlations

Question

Suppose $X$ is a Gaussian random variable and $Y$ is a Gaussian random vector of length $n$ (they are also jointly Gaussian). Let $z$ be a vector with entries $z_i = \frac{\mathrm{Cov}[X, Y_i]}{\sqrt{\mathrm{Var}[X]}}$. Let $A = \mathrm{Var}[Y]$. I am wondering if there is an upper bound for $$\|A^{-1} z\|_2^2.$$ For example, if $A$ is diagonal (i.e., the entries of $Y$ are mutually independent) then $(A^{-1} z)_i^2 = \frac{\mathrm{Cov}[X, Y_i]^2}{\mathrm{Var}[X] \cdot \mathrm{Var}[Y_i]^2} \leq \frac{\mathrm{Cor}[X, Y_i]^2}{\mathrm{Var}[Y_i]} \leq \frac{1}{\mathrm{Var}[Y_i]}$. So in this case $\|A^{-1} z\|_2^2 \leq \frac{n}{\min_i \mathrm{Var}[Y_i]}$. Does a bound like this also exist when the entries of $Y$ are not mutually independent?