  The temperature ($T$)-magnetic field ($H$) phase diagram of YbRh$_2$Si$_2$ in
the vicinity of its quantum critical point is investigated by low-$T$
magnetization measurements. Our analysis reveals that the energy scale
$T^\star(H)$, previously related to the Kondo breakdown and terminating at 0.06
T for $T\to 0$, remains unchanged under pressure, whereas the antiferromagnetic
critical field increases from 0.06 T ($p=0$) to 0.29 T ($p=1.28$ GPa),
resulting in a crossing of $T_N(H)$ and $T^\star(H)$. Our results are very
similar to those on Yb(Rh$_{1-x}$Co$_x$)$_2$Si$_2$, proving that the Co-induced
disorder can not be the reason for the detachment of both scales under chemical
pressure.
