Pantev–Toën-type Fourier–Mukai symmetry conjecture for quasi-BPS categories of Higgs bundles

Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$ with $l=\deg L>2g-2$, and let $B^L$ be the Hitchin base. For rank $r$, numerical invariant $hi$, weight $w$, and spectral-curve genus $g^{\rm{sp}}$, let $\mathbb{T}^L(r,\chi)_w$ denote the corresponding quasi-BPS category, and suppose that $(r,\chi,w)$ satisfies the BPS condition. Over the smooth irreducible spectral-curve locus $(B^L)^{\rm{sm}}$, there is an equivalence $\mathbb{T}^L(r,w+1-g^{\rm{sp}})_{-\chi+1-g^{\rm{sp}}}|_{(B^L)^{\rm{sm}}}\simeq\mathbb{T}^L(r,\chi)_w|_{(B^L)^{\rm{sm}}}$. Pantev–Toën symmetry conjecture. There is a $B^L$-linear equivalence

$$\mathbb{T}^L(r,w+1-g^{\rm{sp}})_{-\chi+1-g^{\rm{sp}}}\simeq\mathbb{T}^L(r,\chi)_w$$

which extends the equivalence over $(B^L)^{\rm{sm}}$, meaning that it commutes with the restriction functors to $(B^L)^{\rm{sm}}$. This conjecture predicts that the Fourier–Mukai symmetry for the relative Picard stacks of smooth spectral curves extends across the full Hitchin base; its resolution status is not specified in the supplied text.