Filliman's extremal projection conjecture for the regular simplex

Let $\Delta_n$ be the regular $n$-simplex in $\mathbb{R}^n$. Fix $1\leq k\leq n$, and let $H_{\ast}$ be a $k$-dimensional subspace such that $T_{\ast}=\operatorname{Proj}_{H_{\ast}}(\Delta_n)$ is a $k$-dimensional simplex. Suppose the vertices of $\Delta_n$ project only onto the vertices of $T_{\ast}$, as evenly as possible: if $w_i$ is the number projecting onto vertex $i$ of $T_{\ast}$, then, writing $n+1=(k+1)\ell+r$ with $r\in\{0,\ldots,k\}$, one has

$$w_i=\begin{cases}\ell+1,&1\leq i\leq r,\\ell,&r<i\leq k+1. \end{cases}$$

Filliman's conjecture. The minimum $k$-dimensional projection volume is attained by $T_{\ast}$:

$$\min_{\substack{H\subset\mathbb{R}^n\dim H=k}}\operatorname{vol}_k(\operatorname{Proj}_H(\Delta_n))=\operatorname{vol}_k(T_{\ast}).$$

Moreover, for $T^{\ast}=\operatorname{Proj}_{H_{\ast}^{\perp}}(\Delta_n)$, the maximum $(n-k)$-dimensional projection volume is attained by $T^{\ast}$:

$$\max_{\substack{H\subset\mathbb{R}^n\dim H=n-k}}\operatorname{vol}_{n-k}(\operatorname{Proj}_H(\Delta_n))=\operatorname{vol}_{n-k}(T^{\ast}).$$

The conjecture proposes the extremal volumes for projections of the regular simplex onto complementary-dimensional subspaces; the stated source does not provide evidence resolving it, so its status remains open.