Square cut swimming trunks should be skin tight whether they are lycra/spandex or nylon. They should have elastic waists/hips and elastic leg holes. It is likely that there is a front lining and a waist cord.
If elastic has not failed then nothing should drop out through slippage. It is possible that the rear of the suit may ride down showing a bit of ass crack (tunnelling) especially if the trunks are worn from the hips rather than the waist but this is not considered a problem these days.
Mini swimming shorts might be mistaken for square cut swimming trunks but be made from non stretch fabric and have no leg elastic. They may or may not have an inner gusset lining to give support. Worn without a gusset liner there would be need of a swim jockstrap to stop slippage and fall out. This may perhaps be what you have.
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