The edge-connectivity conjecture for UBB cardinality

Let $G$ be a graph, and let $\lambda(G)$ denote its edge connectivity, meaning the minimum number of edges whose removal disconnects $G$. For an integer $t$, a $t$-UBB is an uncovering-by-bases structure with parameter $t$, and its cardinality is its number of spanning trees.

Edge-connectivity UBB conjecture. If

$$\lambda(G)=k,$$

then there exists a $(k-1)$-UBB for $G$ with cardinality

$$|E(G)|.$$

The conjecture is motivated by the constructions discussed in the source, where the number of spanning trees in the $t$-UBB is bounded above by the number of edges. The source describes this as possibly optimistic and does not report a resolution.