New answers tagged recessed-lighting
One solution that may be more expensive, but possibly simpler is to create a "trim adapter" specialized to the 4" LED fixtures. The fixtures should mount into the trim and the trim should mount to the existing 6" cans (e.g. via spring holes). Something like this below: The trim adapter can be printed using an online 3D printing service.
Since the ceiling is up already just use standard "old-work" recessed cans. You don't even need "IC" cans since there is no insulation.
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