Finiteness conjecture for odd deficient perfect numbers with a fixed number of prime factors

For a positive integer $k\geq 3$, consider odd deficient perfect numbers whose prime factorization has exactly $k$ distinct prime factors. Finiteness conjecture. For any positive integer $k\geq 3$, there are only finitely many odd deficient perfect numbers with exactly $k$ distinct prime factors. This is a broader finiteness question about the distribution of odd deficient perfect numbers by the number of distinct prime factors, and it remains open in the source.