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Does submodularity property hold for the trace of a positive-definite hermitian matrix?

I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ tr~X^{-1} + tr~(X+A+B)^{-1} \geq tr~(X+A)^{-1} + tr~(X+B)^{-1} $$ hold?

UPD: I have checked it numerically, it appears that it does not hold for the following matrices: \begin{equation} \begin{split} X &= \begin{pmatrix} 0.08151549 & 0.05234424\\ 0.05234424 & 0.17050588 \end{pmatrix} \\ A &= \begin{pmatrix} 0.29185525 & 0.29699319\\ 0.29699319 & 0.30792421 \end{pmatrix} \\ B &= \begin{pmatrix} 0.65213446 & 0.43711443\\ 0.43711443 & 0.2932183 \end{pmatrix} \\ \end{split} \end{equation}

$\newcommand{\tr}{\operatorname{tr}}$Does submodularity property hold for the trace of a positive-definite hermitian matrix?

I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ \tr X^{-1} + \tr(X+A+B)^{-1} \geq \tr(X+A)^{-1} + \tr(X+B)^{-1} $$ hold?

UPD: I have checked it numerically, it appears that it does not hold for the following matrices: \begin{equation} \begin{split} X &= \begin{pmatrix} 0.08151549 & 0.05234424\\ 0.05234424 & 0.17050588 \end{pmatrix} \\ A &= \begin{pmatrix} 0.29185525 & 0.29699319\\ 0.29699319 & 0.30792421 \end{pmatrix} \\ B &= \begin{pmatrix} 0.65213446 & 0.43711443\\ 0.43711443 & 0.2932183 \end{pmatrix} \end{split} \end{equation}

Does submodularity property hold for the trace of a positive-definite hermitian matrix?

I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ tr~X^{-1} + tr~(X+A+B)^{-1} \geq tr~(X+A)^{-1} + tr~(X+B)^{-1} $$ hold?

Does submodularity property hold for the trace of a positive-definite hermitian matrix?

I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ tr~X^{-1} + tr~(X+A+B)^{-1} \geq tr~(X+A)^{-1} + tr~(X+B)^{-1} $$ hold?

UPD: I have checked it numerically, it appears that it does not hold for the following matrices: \begin{equation} \begin{split} X &= \begin{pmatrix} 0.08151549 & 0.05234424\\ 0.05234424 & 0.17050588 \end{pmatrix} \\ A &= \begin{pmatrix} 0.29185525 & 0.29699319\\ 0.29699319 & 0.30792421 \end{pmatrix} \\ B &= \begin{pmatrix} 0.65213446 & 0.43711443\\ 0.43711443 & 0.2932183 \end{pmatrix} \\ \end{split} \end{equation}

Does submodularity property hold for the trace of a positive-definite hermitian matrix?

I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ tr~X^{-1} + tr~(X+A+B)^{-1} \leq tr~(X+A)^{-1} + tr~(X+B)^{-1} $$ hold?

Does submodularity property hold for the trace of a positive-definite hermitian matrix?

I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ tr~X^{-1} + tr~(X+A+B)^{-1} \geq tr~(X+A)^{-1} + tr~(X+B)^{-1} $$ hold?