The BR-linkage conjecture for formats $(2,6,5,1)$ and $(2,5,5,2)$

Let $R$ be a commutative local or graded ring, and let $M$ be a perfect module with a minimal free resolution of one of the formats

$$(2,6,5,1)\quad\text{or}\quad(2,5,5,2).$$

A module is in the BR-linkage class of a Buchsbaum–Rim complex when it can be obtained from such a complex through Buchsbaum–Rim linkage. BR-linkage conjecture. Every perfect module with a free resolution of format $(2,6,5,1)$ or $(2,5,5,2)$ is in the BR-linkage class of a Buchsbaum–Rim complex. This is the module analogue suggested by the preceding ideal-linkage conjectures; the source presents it as an expectation rather than a proved result.