Simple SDP using Hermitian matrices

I would like to run the following semidefinite program $$ \textrm{Minimize}\,\, \textrm{tr}\big({AX}\big)\\ \textrm{subject to}\,\, X \geq B_1\,, X \geq B_2\,, $$ where $A$, $B_1$ and $B_2$ are fixed $2 \times 2$ Hermitian positive-semidefinite matrices and the variable $X$ is also a $2 \times 2$ Hermitian positive-semidefinite matrix. The following is my attempt so far.

MinimizeProblem[A_, B1_, B2_] := Return[SemidefiniteOptimization[Tr[A. X], VectorGreaterEqual[{X - B1, 0}, {"SemidefiniteCone", 2}] && VectorGreaterEqual[{X - B2, 0}, {"SemidefiniteCone", 2}] && VectorGreaterEqual[{X, 0}, {"SemidefiniteCone", 2}], X \[Element] Matrices[{2, 2}, Complexes], {"PrimalMinimumValue", "PrimalMinimizer"}]]; 

This returns no result, but it doesn't give me a specific error either. I suspect the problem comes from using complex matrices, as this seems to be the main way that my problem differs from other SDP examples on this site e.g. MaxCut SDP primal in mathematica, but I'm not sure. In my attempts, I used the following for $A$, $B_1$ and $B_2$, $$A= \begin{pmatrix} 0.7 & 0 \\ 0 & 0.3\\ \end{pmatrix}\,, \qquad B_1= \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1\\ \end{pmatrix}\,, \qquad B_2= \frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1\\ \end{pmatrix}\,.$$ Any help is appreciated.

Edit: If $A$ and $X$ are both positive semi-definite then they can be expanded in their eigenbases as $A=\sum_{i}\lambda_i|u_i \rangle\langle u_i|$ and $X=\sum_{j}\gamma_j|v_j \rangle\langle v_j|$, where every eigenvalue $\lambda_i$ and $\gamma_j$ is real and non-negative. Then $\textrm{tr}\big(AX \big)=\sum_{i, j}\lambda_{i}\gamma_{j}|\langle v_j | u_i \rangle|^2$ is real and non-negative.