Lew's extremal characterization conjecture for simplicial complexes

Let $X$ be a simplicial complex on vertex set $V$ of size $n$, with $h(X)=d$. Write $\mu_k(X)$ for the parameter appearing in the equality below, and let $\Delta_m$ denote the complete simplicial complex on $m+1$ vertices; $\Delta_d^{(d-1)}$ is its $(d-1)$-dimensional skeleton. The symbol $*$ denotes the join of simplicial complexes, and $X\cong Y$ means that $X$ and $Y$ are isomorphic simplicial complexes. Lew's conjecture. If

$$\mu_k(X)=(d+1)(k+1)-dn$$

for some $k$, then

$$X\cong \left(\Delta_d^{(d-1)}\right)^{*(n-k-1)}*\Delta_{(d+1)(k+1)-dn-1}.$$

In particular, $\dim(X)=k$. Lew proposed this as a characterization of the simplicial complexes attaining equality in the relevant spectral-gap lower bound; the supplied source does not state whether the conjecture has been resolved.