Have you ever seen logarithmic burn-in on a Kalman filter before?
The authors cite this work in their response to the review:


  Zhao H, Lu L. Adaptive recursive algorithm with
  logarithmic transformation for nonlinear system identification in
  α-stable noise [J]. Digital Signal Processing, 2015:S1051200415002535.


They take their inspiration from equation (14):


  $J_p(n)=\sum_{i=1}^n \lambda^{n-1} \cdot \log^p (1+|e(i)|)$


It is notable that $e(i)$ is (as one might expect) the error in the signal, and not linearly increasing with the number of steps since the filter was initialized.