A matrix monotonicity question Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane. Consider two positive reals $T_1,T_2>0$ such that $T_1\le T_2$.

My question. Does the following inequality hold true
  $$
\sum_{k\ge 0}e^{A T_1 k} X e^{A^\top T_1 k}\ge \sum_{k\ge 0}e^{A T_2 k} X e^{A^\top T_2 k},
$$
  where $e^\cdot$ denotes the matrix exponential?

I'm able to prove the above inequality in the special case $T_2= m T_1$ where $m$ is an integer, however I have no clue about how to prove (or disprove) it in the general case. Thanks in advance for your help!
 A: The answer is no, in general. Here is a counterexample:
Let
\begin{align*}
    X =
    \begin{pmatrix}
        1 & 0 \\
        0 & 0
    \end{pmatrix},
    \quad \text{and} \quad
    A =
    \begin{pmatrix}
        -1 & -1 \\
         1 & -1
    \end{pmatrix}
    = 
    \begin{pmatrix}
        0 & -1 \\
        1 & 0
    \end{pmatrix}
    - I,
\end{align*}
where $I \in \mathbb{R}^{2 \times 2}$ denotes the identity matrix. Note that the eigenvalues of $A$ are $-1\pm i$, so $A$ is stable. For every $t \in [0,\infty)$ and every $k \in \mathbb{N}_0$ we have
\begin{align*}
    e^{tkA} =
    e^{-kt}
    \begin{pmatrix}
        \cos(kt) & -\sin(kt) \\
        \sin(kt) &  \cos(kt)
    \end{pmatrix}.
\end{align*}
Choose $y = (0,1) \in \mathbb{R}^2$. Then we have
\begin{align*}
   \langle y, e^{tkA} X e^{tkA^T}y\rangle = \langle y, e^{tkA}X(e^{tkA})^T y\rangle = e^{-2kt}\sin^2(kt),
\end{align*}
for all $t \in [0,\infty)$ and $k \in \mathbb{N}_0$, so
\begin{align*}
    \langle y, \sum_{k\ge 0} e^{tkA} X e^{tkA^T} y\rangle = \sum_{k \ge 0} e^{-2kt} \sin^2(kt) =: \varphi(t).
\end{align*}
for all $t \in [0,\infty)$. Now, set $t_1 := \pi$ and $t_2 := \frac{3}{2}\pi$. Then $\varphi(t_1) - \varphi(t_2) = -\varphi(t_2) < 0$, so the matrix
\begin{align*}
    \sum_{k\ge 0} e^{t_1kA} X e^{t_1kA^T} - \sum_{k\ge 0} e^{t_2kA} X e^{t_2kA^T}
\end{align*}
is not positive semi-definite.
