Félix–Oprea–Tanré's generalised toral rank conjecture

Let $F\hookrightarrow E\to B$ be a quasi-nilpotent fibration, with fibre $F$ equal to a torus $T^r$, and suppose that the rational cohomology of $B$ is finite dimensional. Generalised toral rank conjecture. One should have

$$\dim H^*(E;{\mathbb{Q}})\geq 2^r.$$

The source also asks whether, if $F$ is merely rationally elliptic, the weaker inequality $\dim H^*(E;{\mathbb{Q}})\geq\dim H^*(F;{\mathbb{Q}})$ holds. This formulation was proposed by Félix, Oprea, and Tanré; the article provides counterexamples to the generalised claim, so it is refuted.