Card-based protocols utilizing up-down cards with an asymmetrical mark on the face side, allowing it to be distinguished after a 180-degree rotation, have been studied. Previous research has proposed finite-time committed-AND protocols incorporating interlocking operations, which combine shuffling and random rotations of cards. These protocols achieve the theoretical minimum by using only three up-down cards. However, interlocking operations are difficult to implement in practice. In this paper, we prohibit interlocking operations and prove that the minimum number of cards required without such operations is four.