For a Fano threefold $X$, is there a lower bound for $S^3$ where $S\subseteq X$ is an irreducible surface?

Question

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and assume that $-K_X$ is ample, i.e. $X$ is Fano. If $\dim X=2$, then we have that $C^2\geq -1$ for all curves $C\subseteq X$, because $C^2=(K_X+C).C+(-K_X).C\geq -2+1=-1$. Is there a similar statement if $\dim X=3$, i.e. is there a lower bound for $S^3$ where $S\subseteq X$ is an irreducible surface? I can see how to bound expressions like $S^2.T+S.T^2$ for distinct irreducible surfaces $S,T$, but for $S^3$ I have no idea.