time, the value presented in Table \[tablePropKW4\] is valid for epoch JD 2455305.0. For this epoch, which is approximately the mean time of all observed events, a correlation between $P_{\rm orb}$ and $\Delta M_d$ is zero. We also give the time derivatives of the orbital period and the semimajor axis, derived from $\Delta M_d$.

Although the orbit of 1999 KW4 crosses those of Earth, Venus and Mercury, according to JPL HORIZONS system the asteroid experienced only four close approaches to Earth between 2000 and 2019. The approaches took place in May 2001, May 2002, May 2018 and May 2019 at distances of 0.032, 0.089, 0.078 and 0.035 AU, respectively. Since the observed mutual orbital period increase is based on the observations at six effective epochs (apparitions), we can rule out planetary-tug effects as a potential mechanism for the increase.

The uncertainty area of the orbital pole is shown in Fig. \[66391\_LB\_polar\]. The size of the area shrinks with increasing the flattening of the primary $(A_1 B_1)^{1/2}/C_1$. To demonstrate the effect, we constrained the orbital pole uncertainties using three fixed values of the flattening (1.0, 1.2 and 1.4) and plotted the respective areas in the figure.

The uncertainties of the mutual 