The bordering-chamber conjecture for smooth projective toric varieties of Picard number at most four

Let $V$ be a reduced $n\times(n+r)$ $F$-matrix and let $\gamma\in\mathcal{A}_{\Gamma}(V)$ be a non-singular chamber. A chamber is bordering if it lies on the boundary of the relevant chamber configuration. A smooth projective toric variety has Picard number $r$. Bordering-chamber conjecture. If $r\leq 4$, then $\gamma$ is a bordering chamber. Equivalently, every smooth projective toric variety of Picard number $r\leq 4$ admits non-trivial nef divisors that are not big. This would extend the known result for Picard number at most three; the case $r=4$ remains open, while examples for $r\geq 5$ show that the analogous assertion does not hold in general.