Finiteness conjecture for cyclically 4-connected cubic flow roots

Let $G$ be a cyclically 4-connected cubic graph, and let $F(G,t)$ denote its flow polynomial. Let $\tau=(1+\sqrt{5})/2$. Finiteness conjecture. For every $\epsilon>0$, only finitely many cyclically 4-connected cubic graphs have a flow root in

$$(2,\tau^2-\epsilon).$$

This is the flow-polynomial counterpart of the finiteness conjecture for 4-connected plane triangulations. The source presents it as open.