Multiperfect numbers with identical digits

Let g⩾2. A natural number N is called a repdigit in base g if all of the digits in its base g expansion are equal, i.e., if for some m⩾1 and some D∈{1,2,…,g−1}. We call N perfect if σ(N)=2N, where σ denotes the usual sum-of-divisors function. More generally, we call N multiperfect if σ(N) is a proper multiple of N. The second author recently showed that for each fixed g⩾2, there are finitely many repdigit perfect numbers in base g, and that when g=10, the only example is N=6. We prove the same results for repdigit multiperfect numbers.