Weak and strong localization conjectures for hypertoric varieties

Let $K$ be a subtorus of a torus $T$, let $\boldsymbol{\chi}=(\chi_i)$ be the associated characters, and let $\alpha$ be as in the construction of the hypertoric variety. For each circuit $I$, let $W_I$ be the corresponding wall in $X^*(K)$, let $\vec{n}_I$ be a primitive normal vector to $W_I$ in $X_*(K)\subset X_*(T)$, chosen to pair positively with $\alpha$, and set

$$N_I=\sum_i\left|\langle\vec{n}_I,\chi_i\rangle_T\right|,\qquad N=\max_I\{N_I\},$$

with

$$P=\left\{\chi\in X^*(K):\left|\langle\vec{n}_I,\chi\rangle_K\right|\leq N_I\right\}.$$

Here $\lambda=d\Lambda$ is an integral choice of parameter, and $Loc_{\lambda}$ denotes the localization functor. Weak and strong localization conjectures. If $p>N$, then: (i) in the weak form, if $\Lambda\notin P$, the functor $Loc_{\lambda}$ has finite homological dimension; and (ii) in the strong form, when $K$ and $\alpha$ satisfy some positivity condition and the vectors $\vec{n}_I$ are chosen to be positive, it is enough to choose $\lambda$ outside the negative part of $P$ with respect to the orientations given by the $\vec{n}_I$. These conjectures are proposed as the two forms of a localization theorem for hypertoric varieties in positive characteristic, parallel to the localization theorem of Bezrukavnikov–Mirković–Rumynin. The source does not state whether either form is proved or remains open.