Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the positive cone in the topological dual $B'$. Is it true that any (topologically) continuous linear functional which is order-bounded belongs to $K'-K'$?

In other words, let $x^*:B\to \mathbb{R}$ be a linear order bounded functional. Then, being $B$ a lattice, there exists $x^*\lor 0$, which is a linear order bounded functional on $B$. Is it true that $x^*\lor 0$ is topologically continuous if $x^*$ is so?

The main concern is when $K$ is not normal, such as $B=H^1([0,1])$ with the natural norm and ordering given by $K=\{u\geq 0\}$, but I am still not able to find a counterexample in this space.