A strict gap between face and cycle percolation thresholds

Fix integers $D \ge 3$ and $2 \le d < D$, and let $X_\lambda$ be a homogeneous Poisson point process of intensity $\lambda > 0$ in $\mathbb{R}^D$. Consider the Vietoris–Rips complex on $X_\lambda$ whose simplices are the finite subsets with all pairwise Euclidean distances at most $1$, and call two $d$-simplices adjacent when they share a $(d-1)$-face. Let $F_d$ be the event that this adjacency graph on $d$-simplices has an infinite connected component, and $C_d$ the event that there is an infinite, face-connected collection $M$ of $d$-simplices in which every $(d-1)$-simplex is a face of finitely many, and an even number, of members of $M$. Such an $M$ need not represent a nonzero homology class. Put $$\lambda_d^{\mathrm{face}} = \inf\{\lambda > 0 : \mathbb{P}(F_d) > 0\}, \qquad \lambda_d^{\mathrm{cycle}} = \inf\{\lambda > 0 : \mathbb{P}(C_d) > 0\}.$$ Hirsch and Valesin proved $$\lambda_1^{\mathrm{face}} = \lambda_1^{\mathrm{cycle}} \le \lambda_2^{\mathrm{face}} \le \lambda_2^{\mathrm{cycle}} \le \cdots \le \lambda_{D-1}^{\mathrm{face}} \le \lambda_{D-1}^{\mathrm{cycle}} \le \lambda_D^{\mathrm{face}},$$ and that the phase transition for face percolation is sharp. They also proved $\lambda^*_{d,r_0} < \lambda_d^{\mathrm{face}}$ for every $r_0 > 2$, where $\lambda^*_{d,r_0}$ is the threshold of a variant in which two $d$-simplices count as adjacent when their Euclidean distance is at most $r_0$.

Problem. Is $\lambda_d^{\mathrm{face}} < \lambda_d^{\mathrm{cycle}}$ for every $D \ge 3$ and every $2 \le d < D$?