Unimodularity conjecture for super-standard inclusion matrices

Let $n$ be a positive integer. For $0\leq i\leq \frac{n+1}{3}$ and $0\leq j\leq \frac{n+1}{3}$, let $\widetilde{\mathcal{P}}_{i,j}(n)$ be the matrix obtained by stacking the inclusion matrices whose rows are indexed by super-standard $s$-subsets of $\{1,\ldots,n\}$, for $0\leq s\leq i$, and whose columns are indexed by standard $j$-subsets. Here $\mu_i$ and $\mu_j$ denote the corresponding numbers of rows and columns. Unimodularity conjecture. The matrix $\widetilde{\mathcal{P}}_{i,j}(n)$ has dimensions $\mu_i\times\mu_j$, index $1$, and full rank. Consequently, $\widetilde{\mathcal{P}}_{k,k}(n)$ is unimodular for all $n\geq 3k-1$. This conjecture is motivated by computational evidence and the results in the appendix; its resolution is not stated in the source.