between $\tau$ and $D^{-1}$ can be interpreted in the following way that invokes dynamic heterogeneity. As the temperature is lowered, the system consists of many regions with different relaxation time that comes from different local mobilities. We can easily see that the structural relaxation time is dominated by the least mobile regions, that is, by the regions with the longest relaxation time. On the contrary, the average (long time) diffusion characteristics is dominated by the most mobile regions. In other words, the structural relaxation function and the self-diffusion function, respectively, are probing more or less opposite aspects of the relaxation behavior of the system. For an extreme example, one can imagine a system where half of the whole system is frozen (no motion of the component particles) while the remaining half of the system has finite relaxation time with uniformly distributed mobile particles. For this system, the structural relaxation time $\tau$ would be infinite due to the frozen half of system, but inverse of the average diffusion constant $D^{-1}$ is finite due to the mobile part of the system, leading to an extreme breakdown of SE relation. Above simulation result, thus, can be interpreted as an evidence pointing toward the 