The regularity-threshold conjecture for random Steiner complexes

Let $X$ be a random $k$-regular $d$-complex on $n$ vertices, and let $k(d)=4d^2+d+2$ denote the threshold appearing in Theorem 1. The theorem currently assumes $k>k(d)$.

Regularity-threshold conjecture. The condition on $k$ in Theorem 1 can be improved to

$$k>d+1.$$

The bound $k>d+1$ is suggested by the fact that for $k<d+1$, the upper Laplacian $\Delta_{d+1}^+(X)$ has a non-trivial zero eigenvalue and hence the simplicial matrix-tree theorem gives $\kappa_d(X)=0$. The conjecture asks whether the stronger regularity hypothesis arising from Garland's method can be reduced to this natural threshold.