Halpern-Leistner's spanning conjecture for Fano varieties

Let $X$ be a Fano variety, let $\tau\in\mathrm{H}^2(X)$, and let $\mathscr{S}\subset\mathbf{C}^*$ be a sector. Let $\sigma(\cE_\tau)$ be the spectrum of the Euler operator $\cE_\tau\star_\tau(-)$, and write $\lvert\sigma(\cE_\tau)\rvert$ for its underlying set. For a real number $r$, define

$$F^r\mathrm{H}_{\mathrm{alg}}^\bullet(X):=\left\{\alpha\in\mathrm{H}_{\mathrm{alg}}^\bullet(X):\log\lvert\cZ_{te^{\mathtt{i}\varphi}}^\tau(\alpha)\rvert\leq rt^{-1}+o(t^{-1})\text{ as }t\to0\right\}.$$

Halpern-Leistner's spanning conjecture. For any Fano variety $X$, there exist $\tau\in\mathrm{H}^2(X)$ and a sector $\mathscr{S}\subset\mathbf{C}^*$ such that, for every $\tau$-admissible phase $\varphi$ with $\mathbf{R}_{>0}e^{\mathtt{i}\varphi}\subset\mathscr{S}$, there is a quasi-convergent path $\sigma_{t,\varphi}^\tau$ in $\operatorname{Stab}(X)$ for which, for every $r=\operatorname{Re}(-\lambda e^{-\mathtt{i}\varphi})$ with $\lambda\in\lvert\sigma(\cE_\tau)\rvert$, the space $F^r\mathrm{H}_{\mathrm{alg}}^\bullet(X)$ is spanned by Chern characters of limit semistable objects of $\sigma_{t,\varphi}^\tau$.

This conjecture is an interpretation of Halpern-Leistner's proposal connecting asymptotic quantum-connection growth with categorical decompositions. The asserted spanning property is open.