Differences between the KF and an LTI Filter

Solving the Discrete Riccati Equation for $t\to \infty$ we can find the asymptotic values of $P_{\infty }$ and $K_{\infty }$

KF:

LTI Filter:

So both the error and the covariance go to zero, but for the $K$ requirements to find a fast solution are conflicting: