Periodicity of cubic vectors without Preliminary Stage 2

The sin-squared algorithm is an algorithm with a Preliminary Stage 2 followed by a main stage, and a vector is cubic when its coordinates generate a cubic number field. Periodicity means that the algorithm eventually repeats its state.

Periodicity conjecture without Preliminary Stage 2. Excluding Preliminary Stage 2 from the sin-squared algorithm will not affect periodicity for cubic vectors.

The conjecture is motivated by computed examples in which iterations of the main stage applied to a supporting basis reach a separating basis in finitely many steps without requiring Preliminary Stage 2. The supplied text gives no resolution, so its status remains open.