Fano–Enriques threefolds of big genus; the case of conic bundle

A pair $(U,A)$ is called a Fano–Enriques threefold if $U$ is a projective normal threefold and $A$ is an ample effective Cartier divisor on $U$ and $A$ is a smooth Enriques surface. A natural discrete of a pair is an integer number $g = A^3/2 + 1$ called genus of $(U,A)$. Prokhorov suggested a method to classify such threefolds. In particular he proved a sharp bound $g \leqslant 17$.
We show that that there is better bound: either $g = 17$ or $g \leqslant 13$. Log-minimal model program reduces studying of $U$ to studying a Mori fiber space. We discuss the most interesting case of conic bundle.