The improved bound for big and nef tautological bundles on Enriques surfaces

Let $F$ be a $(k-1)$-very ample bundle of rank $r$ on an Enriques surface. Write

$$\delta=\frac{1}{2}+\frac{1}{2}\langle v,v\rangle=rc_2-\frac{r-1}{2}c_1^2-\frac{r^2-1}{2},$$

where $v=v(F)=\operatorname{ch}(F)\sqrt{\operatorname{Td}(X)}$ is the Mukai vector. Improved bound conjecture. For all ranks, odd or even, if

$$\chi(F)\geq \left(\frac{5r}{4}+2\right)k,\qquad \delta\geq 0,$$

then $F^{[k]}$ is big and nef. The preceding theorem establishes the analogous assertion for odd rank under the stronger hypothesis $\chi(F)\geq 2k(r+1)$; the proposed weaker bound is intended to extend the result to all ranks, while its general validity remains open.