Nonminimal absorption law

Determine whether the local absorption moves (1,a)→(a+1) and (a,1)→(a+1) induce homology equivalences whenever d>a+1, and whether (1,a,1)→(a+2) induces a homology equivalence whenever d>a+2 precisely when a is even or d is not congruent to 1 modulo a+1.

Background

The paper disproves unrestricted homology invariance for the move (1,a,1)→(a+2) using a=3 and d=9. Computations suggest that the failures may follow a precise parity and congruence pattern away from the minimal admissible degree.

The conjecture formulates the proposed exact conditions for all three local moves. The available evidence is limited to finite computational ranges.