Alavi–Erdős–Malde–Schwenk conjecture on unimodality for trees and forests

A tree is a connected acyclic graph, and a forest is an acyclic graph. For a graph $G$, its independence sequence $(i_0(G),i_1(G),\ldots,i_{\alpha(G)}(G))$ records the numbers of independent sets of each cardinality. Alavi–Erdős–Malde–Schwenk conjecture. Every tree and every forest has a unimodal independence sequence. The paper states that this conjecture remains open, in contrast with the disproved unimodality conjecture for all well-covered graphs.