Exceptional-sheaf version of the Bogomolov–Gieseker inequality

Let $X$ be the canonical line bundle of $\mathbb{P}^3$, with projection $\pi:X\to\mathbb{P}^3$. Let $\beta\in\mathbb{R}$ and let $\mathcal{E}$ be an exceptional locally free sheaf of finite rank on $\mathbb{P}^3$. Let $\mu_1(\mathcal{E})$, $\mu_2(\mathcal{E})$, $\alpha_{\mathcal{E}}^{\beta}$, $v_i^{\beta}$, and the hearts and tilt slope appearing below be as defined in the source. Exceptional-sheaf Bogomolov–Gieseker conjecture. (1) If $\beta<\mu_1(\mathcal{E})$ and $i_*\mathcal{E}\in\mathrm{Coh}_{0}^{\beta}(X)$ is $\nu^{\beta,\alpha_{\mathcal{E}}^{\beta}}$-stable, then there exists $a_0<\frac{v_3^{\beta}(i_*\mathcal{E})}{v_1^{\beta}(i_*\mathcal{E})}$ such that every $\nu^{\beta,\alpha_{\mathcal{E}}^{\beta}}$-semistable object $E\in\mathrm{Coh}_{0}^{\beta}(X)$ with tilt-slope $\beta$ and $\mathrm{Hom}_{\mathrm{Coh}_{0}^{\beta}(X)}(i_*\mathcal{E},E)=0$ satisfies

$$v_3^{\beta}(E)\leqslant a_0v_1^{\beta}(E).$$

(2) If $\beta>\mu_2(\mathcal{E})$ and $i_*\mathcal{E}[1]\in\mathrm{Coh}_{0}^{\beta}(X)$ is $\nu^{\beta,\alpha_{\mathcal{E}}^{\beta}}$-stable, then there exists $a_0<\frac{v_3^{\beta}(i_*\mathcal{E})}{v_1^{\beta}(i_*\mathcal{E})}$ such that every such $E$ with $\mathrm{Hom}_{\mathrm{Coh}_{0}^{\beta}(X)}(i_*\mathcal{E}[1],E)=0$ satisfies the same inequality. This is proposed as an analogue of the main Bogomolov–Gieseker conjecture after removing the exceptional object from consideration; no resolution is given.