Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =x) = \begin{cases} &1-\alpha, &x=0 \\ &\alpha, &x=1 \end{cases}$.

Now my problem is given a random vector $\boldsymbol{X}$, and $\text{supp}\boldsymbol{X} = [0,1]^n$, and $\mathsf{E}\boldsymbol{X}=\boldsymbol{\alpha}$. After a linear transformation $\boldsymbol{H} (\boldsymbol{H} \succ \boldsymbol{0})$, I want to know whether the maximum trace of the covaraince matrix $\text{cov}(\boldsymbol{H}\boldsymbol{X})=\mathsf{E}[(\boldsymbol{H}\boldsymbol{X}-\mathsf{E}[\boldsymbol{H}\boldsymbol{X}])(\boldsymbol{H}\boldsymbol{X}-\mathsf{E}[\boldsymbol{H}\boldsymbol{X})^\text{T}]$ can be achived by a discrete random vector whose support $\text{supp}\boldsymbol{X}=\{0,1\}^n$.

The trace can be expanded as $\text{tr}(\text{cov}(\boldsymbol{H}\boldsymbol{X}))=\sum_{k=1}^{n} h_{i,k}^2 \mathsf{E}{\bigl(X_k-\mathsf{E}{X_k}\bigr)^2} + \sum_{k=1}^{n} \sum_{\substack{\ell=1\\\ell\neq k}}^{n} h_{i,k} h_{i,\ell} \bigl( \mathsf{E}{X_{k} X_{\ell}} - \mathsf{E}{X_{k}}\mathsf{E}{X_{\ell}}\bigr)$. If we use the similar method as in the random variable case, we can maximize the first term of RHS of above equation, be the change of second term of RHS cannot be determined.