The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
MAT, BENCH, KETTLEBELL, and ROPE all fit the category "Things in a gym". Training rooms are their habitat. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Soft things
VELVET / PILLOW / MIST / FLEECE
VELVET, PILLOW, MIST, and FLEECE all fit the category "Soft things". They suggest a gentle texture. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things you fold
LETTER / LAUNDRY / ORIGAMI / MAP
LETTER, LAUNDRY, ORIGAMI, and MAP all fit the category "Things you fold". A crease is part of the process. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Kitchen verbs
FOLD / WHISK / SEAR / DICE
FOLD, WHISK, SEAR, and DICE all fit the category "Kitchen verbs". These are actions from a recipe. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.