The singular Bayer–Macrì–Toda conjecture

Let $X$ be a $3$-dimensional projective scheme over a field $d534$ which is lci in codimension $2$, and let $H$ be an ample divisor. Assume that the Chern characters $d539_i$ are defined for $0d539id5392$, and that $X$ is either lci or $d4d9$-factorial and normal, so that $d539_3(E)$ is defined for $Ed539d539^b_H(X)$. Let $d539_{b,w}$ be the tilt slope associated with $Z^{b,w}$, and let $53f_{X,H}$ be the relevant Bogomolov–Gieseker function. Take $534d5390$ such that

$$53f_{X,H}(x)d539\frac{1}{2}x^2+534.$$

Bayer–Macrì–Toda conjecture. There exists $546d53953c(X)_{4d9}$ with $546.Hd5390$ such that, for every $w>\frac{1}{2}b^2+534$ and every $539_{b,w}$-semistable object $Ed539d539^b_H(X)$, the quadratic Bogomolov–Gieseker-type inequality described in Remark 2.4 holds, involving $539_i(E).H^{3-i}$ and $546.539_1(E)$. The coefficients depend on $546.H$, $(b,w)$, and $534$. This conjecture is the singular analogue of the BMT conjecture and extends the cited conjecture and question; its precise inequality is given in the source context rather than in the candidate span.