Takahashi's vanishing conjecture for DHKK complexities in singularity categories

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{D_{sg}}(R)$ the singularity category of $R$, defined as the Verdier quotient of the bounded derived category of finitely generated $R$-modules by perfect complexes. Let $G$ be a generator of $\operatorname{D_{sg}}(R)$, meaning that its thick closure is $\operatorname{D_{sg}}(R)$, and let $X$ be any object of $\operatorname{D_{sg}}(R)$. Takahashi's conjecture. One has

$$\delta_t(G,X)=0$$

for all nonzero real numbers $t$. The conjecture concerns the vanishing of the DHKK complexity in singularity categories; the paper studies this vanishing and proves boundedness results for the set of real numbers where the complexity does not vanish in various cases, while the full assertion remains unresolved in the stated generality.