Exact formula for the spin geography threshold

Prove that for every even integer s\geq 0, the threshold defining the \infty^2-property satisfies \Lambda_s=|8s-1|.

Background

The quantity \Lambda_s is the smallest positive odd integer such that sE_8\oplus qH has the \infty^2-property for every odd q\geq\Lambda_s. The paper proves conditional lower bounds and asymptotic upper bounds for this threshold.

Motivated by these results, the authors formulate an exact conjecture. They explicitly state that the conjecture remains unresolved for every value of s; for example, it predicts \Lambda_2=15, while the best cited upper bound is \Lambda_2\leq155.