The maximal independent set (MIS) problem is one of the most fundamental problems in the field of distributed computing. This paper focuses on the MIS problem with unreliable communication between processes in the system. We propose a relaxed notion of MIS, named almost MIS (ALMIS), and show that the loosely-stabilizing algorithm proposed in our previous work can achieve exponentially long holding time with logarithmic convergence time and space complexity regarding ALMIS, which cannot be achieved at the same time regarding MIS in our previous work.