Conjecture on divisors parameterized by stable objects on singular cubic surfaces

Let $Y$ be a smooth cubic threefold, let $S$ be a singular hyperplane section, and let $D$ be a Weil divisor on $S$ such that $\mathcal{O}_S(D)$ is a stable object parameterized by $\mathcal{M}_{\sigma}(w)$. Suppose the off-diagonal part of $\widetilde{H(Y)}$ parameterizes type (I) and (III) schemes with an order, whose supports are intended to determine $D$. Divisor-support conjecture. The divisor $D$ has the form $L_1-L_2$, where $L_1,L_2$ are two lines on $S$, possibly singular, and one of the following holds: (i) $L_1$ and $L_2$ are disjoint, though the lines may pass through singularities; or (ii) $L_1$ and $L_2$ intersect at one point $p$, which is a singularity of $S$. The explicit expression of $D$ on singular hyperplane sections is not otherwise known, so this conjecture would identify the divisors arising from the stable moduli space.