Banach spaces which always produce octahedral spaces of operators

We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is octahedral for every non-zero Banach space $Y$. They are those satisfying that, for every finite dimensional subspace $Z$, ${\mathrm{\ell }}_{\mathrm{\infty }}$ can be finitely-representable in a part of $X$ kind of ${\mathrm{\ell }}_1$-orthogonal to $Z$. We also prove that $L(Y,X)$ is octahedral for every $Y$ if, and only if, $L({\mathrm{\ell }}_p^n,X)$ is octahedral for every $n\in \mathbb{N}$ and $1<p<\mathrm{\infty }$. Finally, we find examples of Banach spaces satisfying the above conditions like $\text{Lip}(M)$ spaces with octahedral norms or $L_1$-preduals with the Daugavet property.