Linear lower bound for smallest facets of Vietoris–Rips complexes of hypercubes

Let $s(n;r)$ denote the size of the smallest facet in $\operatorname{\mathcal{VR}}(Q_n;r)$. The source proves $s(n;r)\leq2r$ when a Hadamard matrix of order $2r$ exists and $n\geq2r-1$. Smallest-facet lower-bound conjecture. There is a universal constant $c$ such that

$$s(n;r)\geq cr$$

for all $n$ sufficiently large compared with $r$. Establishing this would provide a linear lower bound matching the scale of the known upper bound up to a universal constant; the conjecture remains open.