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!