Layer-width conjecture for ideality in multilayer random networks

Consider a network with $n$ layers, with $L_k$ agents in layer $k$, independent random connections between consecutive layers, and each connection equally probable. Let $\hat{s}_\text{ideal}$ denote the ideal final estimate. As the total number of agents increases, suppose the layer sizes satisfy

$$L_k\leq L_{k+1}\quad\text{for }1\leq k<n-1.$$

Multilayer random-network conjecture. Under these conditions,

$$P(\hat{s}=\hat{s}_\text{ideal})\to 1.$$

Conversely, if $L_1>L_k$ for some $1<k<n$, then

$$P(\hat{s}=\hat{s}_\text{ideal})\to 0.$$

The conjecture extends the paper's three-layer random-network result to more than three layers. Simulations for four-layer networks are reported as supporting evidence, while a general proof is not provided.