Minimal wildcard density for NP-hardness of rank-2 wildcard decomposition

Ascertain the asymptotically minimal number of wildcard entries in a tensor necessary to ensure that deciding the existence of a rank-2 decomposition over a fixed finite field (e.g., Z/2Z) remains NP-hard.

Background

The paper shows that rank-2 tensor decomposition with wildcard entries over Z/2Z is NP-hard via a reduction from NAE-3SAT, using a construction with many wildcards. It remains open how few wildcard entries suffice to retain NP-hardness, which would refine the hardness landscape for partially specified tensors.