Pointwise minimal decomposition conjecture for unital tensors with covering systems

Let $P_{abc}$ be a unital tensor that admits a covering system of partial algebras. A pointwise decomposition is a positive tensor rank decomposition

$$P_{abc}=\sum_{i=1}^R \beta_i p_a^i p_b^i p_c^i$$

such that every factor $p^i$ corresponds to a potential homomorphism, and the decomposition is minimal when its tensor rank cannot be reduced. Pointwise minimal decomposition conjecture. A minimal positive tensor rank decomposition of $P_{abc}$ is a pointwise decomposition. The source presents this as suggested by the exact-circle example and does not provide a proof, so it remains open.