A complete proof of the De Vylder and Goovaerts conjecture for homogeneous risk models

De Vylder and Goovaerts (2000) conjectured that the finite-time ruin probability in a homogeneous risk model is greater than or equal to the corresponding ruin probability in an associated model with equalized claim amounts. This conjecture holds provided that the conjecture asserting that the same inequality holds for the conditional finite-time ruin probabilities, conditioned on n claims occurring during the finite time, for all n ≥ 1, is true. They proved the conjecture for n=1 and n=2, but left the case n ≥ 3 as an open problem. Kim et al. (2021) resolved the case n=3. In this paper, we completely resolve the conjecture for all n.