Gröbner basis property for tensor-train defining minors

Prove that, for tensor-train varieties of arbitrary order, the minors defining the tensor-train ideal form a Gröbner basis with respect to the diagonal term order specified in Definition 5.2, thereby establishing the conjectured Gröbner basis property beyond order three.

Background

The paper establishes that the prime ideal of a tensor-train variety is generated by the minors of its consecutive flattening matrices. It then asks whether this generating set is also a Gröbner basis for a suitable term order. The authors define a global diagonal term order that is diagonal on every flattening and prove the desired Gröbner basis statement for order-3 tensor trains.

For tensor trains of arbitrary order, the Gröbner basis assertion remains unresolved. The paper notes that existing proofs for order three do not straightforwardly generalize, including a crucial intermediate result from the cited literature that fails under its obvious generalization. Establishing the conjecture would provide a uniform Gröbner basis description for the defining ideals of all tensor-train varieties.