Cohomological lower-bound conjecture for non-Fano hypersurfaces

Let $k$ be an algebraically closed field and let $i\colon X=V(f)\hookrightarrow\mathbb{P}^n_k$ be a closed embedding of a smooth irreducible hypersurface of degree $d$. For a bounded complex $\mathcal C$ of coherent sheaves on $X$, define

$$\rho(\mathcal C)=\sum_{r=0}^n h(\mathbb{P}^n_k,i_*(\mathcal C)\otimes\Omega_{\mathbb{P}^n_k}^r(r)),$$

where $h(Y,\mathcal C)=\sum_{j\in\mathbb{Z}}\dim_k\mathbf{R}^j\Gamma(Y,\mathcal C)$. Put $a\coloneqq n+1-d$. Cohomological lower-bound conjecture. If $a\leq0$, equivalently if $X$ is not Fano, then for any nonzero object $\mathcal C\in\operatorname{D}^{\operatorname{b}}(X)$ one has

$$\rho(\mathcal C)\geq 2^{e+1},\qquad e\coloneqq\left\lfloor\frac{n}{2}\right\rfloor.$$

The conjecture proposes a uniform lower bound on the cohomological size of every nonzero derived-category object on a non-Fano smooth hypersurface; the source states that the assertion is false without the assumption $a\leq0$.