Non-invertible Conway defects conjecture for K3 sigma models

Let $\Pi^{\natural}\subset \Lambda\otimes\mathbb{R}\subset V^{f\natural}_{tw}$ be a four-dimensional subspace, and let $\mathsf{Top}_{\Pi^{\natural}}$ be the full subcategory of topological defects defined by the condition in equation (TopPi). Let $\mathsf{Top}^{K3}_{\mathcal{C}}$ denote the tensor category of topological defects of a K3 nonlinear sigma model $\mathcal{C}$ preserving $\mathcal{N}=(4,4)$ and spectral flow. Non-invertible Conway defects conjecture. There exists a nonlinear sigma model $\mathcal{C}$ on K3 such that (i) there is an equivalence of tensor categories

$$F:\mathsf{Top}_{\Pi^{\natural}}\longrightarrow\mathsf{Top}^{K3}_{\mathcal{C}};$$

(ii) for every $\mathcal{L}\in\mathsf{Top}_{\Pi^{\natural}}$, the twining genera coincide,

$$\phi^{\mathcal{L}}(V^{f\natural},\tau,z)=\phi^{F(\mathcal{L})}(\mathcal{C},\tau,z);$$

and (iii) there is an isometry of Hilbert spaces

$$\varphi:V^{f\natural}_{tw}(1/2)\stackrel{\cong}{\longrightarrow}\mathcal{H}^{K3}_{\mathrm{RR},gr}$$

intertwining the defect actions:

$$\varphi\circ \widehat{\mathcal{L}}_{\rvert V^{f\natural}_{tw}(1/2)}=\widehat{F(\mathcal{L})}_{\rvert \mathcal{H}^{K3}_{\mathrm{RR},gr}}\circ\varphi.$$

The conjecture proposes that the Conway super-vertex operator algebra captures not only invertible K3 symmetries but also the corresponding non-invertible topological defects and their action on the Ramond ground-state Hilbert space.