Decomposition of weak* convergent nets into positive weak* convergent nets

Question

Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^*$ ordered by the dual cone. Let $E\subset F^*$ be a closed subspace that separates points of $F$ and such that the dual cone intersected with $E$ is spanning for $E$. Consider the initial topology $\sigma(E,F)$ induced on $E$ by $F$.

Given a net $\omega_\lambda\in E$ converging to $0$ in $\sigma(E,F)$, do there exist nets $0\le \psi_\lambda, \varphi_\lambda$ of positive elements converging to $0$ in $\sigma(E,F)$ such that $\omega_\lambda = \psi_\lambda - \varphi_\lambda$? If not in this generality, are there further conditions on $E$ under which one can say more, e.g. $E$ being a Banach lattice/AL space?

For example if $E$ is base normed by the base $B = \{\omega\in E\ |\ \omega\ge 0, \omega(e) = 1\}$, the statement holds true if $\sigma(E,F)$ is replaced by the norm topology. That is, every net converging to $0$ in norm can be decomposed into positive nets converging to $0$ in norm. Is the weak version also true in this setting?

More concretely, I suspect the weak version to be true at least for $F = \ell^\infty(\mathbb N, \mathbb R)$ and $E=\ell^1(\mathbb N, \mathbb R)$ with the usual order and decomposition into positive and negative part of a sequence. However, even in this case I don't have a proof nor a counterexample so far and would appreciate either.

By order unit Banach space I mean a real Banach space that is ordered by a closed and pointed cone (that is the cone does not contain any non-trivial subspace). Furthermore there shall exist an element $e$ with the property that for every $x\in F$ there is $\lambda \ge 0$ with $\lambda e\ge x\ge -\lambda e$ and such that $\|x\|$ is the infimum over all such $\lambda$. The dual cone consists of the positive linear functionals on $F$. By base normed by the base $B$ I mean that every positive element $\omega\ge 0$ of $E$ is contained in exactly one set of the form $\lambda B$ where $\lambda\ge0$ and with the property that $$\forall_{\omega\in E}: \|\omega\| = \inf_{\substack{\psi,\varphi\ge 0\\\omega = \psi - \varphi}} \psi(e) + \varphi(e).$$

Answer 1

Based on the comment by @BillJohnson: If $0\le \omega\in E$ then $\|\omega\| = \omega(e)$ implying that a net of positive elements in $E$ converges to $0$ in weak$^*$ topology if and only if it converges to $0$ in norm. Thus, a positive answer to the question would imply that a net (not necessarily positive) converges to $0$ in norm if and only if it converges weak$^*$ to $0$, which is clearly false. Explicitly, if $E$ is infinite-dimensional then the unit sphere of $E$ is weak$^*$ dense in the unit ball of $E$. Hence, there exists a net $\omega_\lambda$ with $\|\omega_\lambda\| = 1$ but converging to $0$ weak$^*$, so the question has a positive answer if and only if $E$ is finite-dimensional.