Coboundary expansion conjecture for random balanced Cayley complexes

Let $G$ be a finite group, let $A\subseteq G$, and let $Y_{A,k}$ denote the random balanced Cayley complex associated with $A$. For a binary $k$-cochain $\phi\in C^k(X;\mathbb{F}_2)$, write

$$\|\phi\|_{\rm H}=\left|\left\{\sigma\in X(k):\phi(\sigma)\neq 0\right\}\right|$$

for its Hamming norm, and

$$\|\phi\|_{\rm csy}=\min\left\{|\operatorname{supp}(\phi+d_{k-1}\psi)|:\psi\in C^{k-1}(X;\mathbb{F}_2)\right\}$$

for its cosystolic norm. Define the $k$-th coboundary expansion constant by

$$h_k(X)=\min\left\{\frac{\|d_k\phi\|_{\rm H}}{\|\phi\|_{\rm csy}}:\phi\in C^k(X;\mathbb{F}_2)\setminus B^k(X;\mathbb{F}_2)\right\}.$$

Coboundary expansion conjecture for random balanced Cayley complexes. For any fixed $k\geq 1$ there exist constants $C(k)<\infty$ and $\epsilon(k)>0$ such that for any group $G$, the random balanced Cayley complexes $Y_{A,k}$ with $|A|=C(k)\log D(G)$ satisfy $h_{k-1}(Y_{A,k})\geq\epsilon(k)$ a.a.s. as $|G|\to\infty$.

This would provide a coboundary-expansion analogue of the paper's logarithmic-size result for the spectral gap, establishing robust cohomological triviality for random balanced Cayley complexes. The source presents it as a suggested conjecture and gives no resolution.