Vigué-Poirrier conjecture on exponential growth of free-loop-space homology

Let $X$ be a simply connected space and let $\mathcal{L}X=\operatorname{Map}(S^1,X)$ be its free loop space. Write $\Omega X$ for the based loop space, and call $X$ rationally hyperbolic if $\dim(\pi_*(X)\otimes\varmathbb{Q})=\infty$. Vigué-Poirrier's conjecture. If $X$ is rationally hyperbolic, then $H_*(\mathcal{L}X;\varmathbb{Q})$ grows exponentially. This conjecture concerns the relationship between rational homotopy growth and the homology of free loop spaces; the paper presents new examples supporting it, while the general assertion remains open.