Is a minimal Cauchy filter bounded?

Question

Let $E$ be a $K$-vector space and $A, B$ two subsets of $E$.

1. We say $A$ absorbs $B$ if there is a $\alpha>0$ such that $B \subseteq \lambda A$ for all $\lambda\in K$ such that $\lambda \geq \alpha$.

2. A set $B$ is bounded if it is absorbed by every neighborhood of zero.

3. A filter $\mathfrak{F}$ is bounded if there exists a bounded set $X\in \mathfrak{F}$.

4. A filter $\mathfrak{F}$ on a set $A$ is a Cauchy filter if for every neighborhood $V$ of zero, there is a set $X\in \mathfrak{F}$ such that $X - X\subseteq V$, that is, $x-y\in V$ for all $x,y\in X$.

5. A Cauchy filter $\mathfrak{F}$ is minimal if there exists no Cauchy filter coarser than $\mathfrak{F}$ and distinct from $\mathfrak{F}$. Given every Cauchy filter $\mathfrak{F}$, there exists a unique minimal Cauchy filter $\mathfrak{F}_0$ coarser then $\mathfrak{F}$.

Is a minimal Cauchy filter bounded? If it is, could you give me a reference or reason.