Halpern-Leistner's determinant-dualizing-complex conjecture

Let $f:X\to Y$ be a morphism between quasi-smooth, locally almost of finite presentation derived Artin stacks. Let $L_{X/Y}$ be the relative cotangent complex and let $\omega_{X/Y}$ be the relative dualizing complex. Halpern-Leistner's conjecture. There is a canonical isomorphism

$$\omega_{X/Y}\cong \det(L_{X/Y})[\operatorname{rank}(L_{X/Y})].$$

Halpern-Leistner proved a weak version in which the two sides become isomorphic after restriction to $\pi_0(X)$. The full canonical isomorphism is therefore not established by the supplied text.