Although it causes an Equipment to become attached to a creature, reconfigure is not an "equip ability" for the purpose of cards like Fighter Class and Leonin Shikari.
Lion Sash
An Equipment doesn't become tapped when the permanent it's attached to becomes tapped. For example, if you attack with a creature that is equipped with Acquisition Octopus, then use reconfigure to unattach Acquisition Octopus after combat, the Octopus will be untapped and could be used to block during your opponent's turn.
An Equipment creature with reconfigure can be attached to creatures by effects other than its reconfigure ability, such as the activated ability of Brass Squire.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Yongjae Choi
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