Joyce–Safronov conjecture on matrix factorizations for shifted Lagrangians

Let $U$ be a smooth $\mathbb{C}$-scheme and let $\phi\colon U\to\mathbb{C}$ be a regular function. Let $\mathbf{M}\to\mathbf{Crit}(\phi)$ be a $(-1)$-shifted Lagrangian such that $\operatorname{vdim}\mathbf{M}-\dim U$ is even, equipped with orientation data and a spin structure. Joyce–Safronov conjecture. There exists an object $\mu_{\mathbf{M}}\in\operatorname{MF}(U,\phi)$ in the matrix factorization category $\operatorname{MF}(U,\phi)$, associated with $\mathbf{M}$, with prescribed local behavior as described by Joyce and Safronov. The conjecture proposes a categorical, or Fukaya-type, assignment of objects in matrix factorization categories to shifted Lagrangians; the paper establishes a $K$-theoretic version of the associated Joyce–Safronov statement, while the asserted object-level construction is the conjectural input.