Conjecture on maximal vertex numbers of maxout polytopes

A $(d,n,m)$-maxout polytope is a polytope specified by the parameters $d$, $n$, and $m$, and $f_0(P)$ denotes the number of vertices of a polytope $P$. Maxout polytopes vertex-number conjecture.

1. The maximal number of vertices of a $(3,n,1)$-maxout polytope equals

$$4\sum_{k=0}^{2}\binom{n-1}{k}$$

if $n$ is odd, and equals

$$4\sum_{k=0}^{2}\binom{n-1}{k}-(n-2)$$

if $n$ is even.

2. For $4\leq d\leq n$, the maximal number of vertices of a $(d,n,1)$-maxout polytope equals

$$4\sum_{k=0}^{d-1}\binom{n-1}{k}.$$

3. For $2\leq d\leq n$, the maximal number of vertices of a $(d,n,m)$-maxout polytope equals

$$2\sum_{k=0}^{d-1}\binom{m-1}{k}\cdot\max\left\{f_0(P)-2\sum_{k=0}^{d-1}\binom{n-1}{k}:P\text{ is a }(d,n,1)\text{-maxout polytope}\right\}.$$

These claims arise from computational bounds and examples for three-dimensional zonoboxtopes. The source provides no resolution evidence, so the conjecture remains open.