We show how many of the congruences between Ikeda lifts and non-Ikeda lifts, proved by Katsurada, can be reduced to congruences involving only forms of genus 1 and 2, using various liftings predicted by Arthur's multiplicity conjecture. Similarly, we show that conjectured congruences between Ikeda–Miyawaki lifts and non-lifts can often be reduced to congruences involving only forms of genus 1, 2 and 3.
Neil DUMMIGAN. "Lifting puzzles and congruences of Ikeda and Ikeda–Miyawaki lifts." J. Math. Soc. Japan 69 (2) 801 - 818, April, 2017. https://doi.org/10.2969/jmsj/06920801