Largest rigid facet conjecture for Vietoris–Rips complexes of hypercubes

Let $Q_n=\\{\pm1\\}^n$ be the Hamming cube with Hamming distance. A subset is maximally diameter-$r$ if its diameter is $r$ and adjoining any point outside it produces diameter greater than $r$; it is rigid if every point has a point at distance equal to the diameter. Let $b'(n;r)$ be the size of the largest maximally diameter-$r$, rigid subset of $Q_n$, equivalently the largest such rigid facet in $\operatorname{\mathcal{VR}}(Q_n;r)$. Largest rigid facet conjecture.

$$b'(n;r)=2^r$$

whenever $n\geq r$. The subcube $Q_r\times\\{\mathbf 1_{n-r}\\}$ supplies an extremal candidate of size $2^r$, but the conjectured upper bound is not proved in the source.