is equivalent to $({{\mathbb}{F}}_n, \sigma)$, then there is a very free curve with smaller $-K_X$ degree.

Let $D$ be the unique (possibly reducible) section such that $D \cdot D=-n, (n \geq 0)$. Write $C=D+cF$. Then $c \geq n$ since $c-n=C \cdot D \geq 0$. In addition, $$a=C \cdot C=2c-n \geq c,$$ and $$2c \geq 2c-n=C \cdot C =a.$$ On the other hand, we have $$2c \leq c(-K_X \cdot F) \leq -K_X \cdot C=a+4.$$ Note that $-K_X \cdot F \geq 2$ since a general fiber $F$ is a free curve. Thus we have the following possibilities.

1.  $a=3, c=3$. Then $-K_X \cdot F=2$ and $-K_X \cdot D=1$. This is impossible since the pseudo-index of the Fano variety $X$ is assumed to be at least $2$.

2.  $a=3, c=2$. Since $-K_X \cdot D \geq 2$, $-K_X \cdot F$ is not $3$. Then $-K_X \cdot F=2$ and $n=1, -K_X \cdot D=3$.

3.  $a=4, c=4$. Then $-K_X \cdot F=2$. But this would imply that there are two free curves in $X$ with $-K_X$ degree $2$, each passing through a very general point, intersecting each other. This is impossible if the two points are chosen to be very general.

4.  $a=4, c=3$. Then $-K_X \cdot 