Domination conjecture for Turán hypergraphs

Let $[n]=\{1,\ldots,n\}$, and let $C(n)$ be the family

$$C(n):=\{abc:\ a+c\leq n,\ 2a+b\leq n,\ 1\leq a<b<c\leq n\}.$$

For a $3$-uniform hypergraph $H$, say that $H$ dominates $C(n)$ when the corresponding rows and columns of the third compound matrix have full row rank, as in the paper's definition of domination. A domination conjecture for Turán hypergraphs. If $H$ is a $3$-uniform hypergraph on $[n]$ such that every four vertices span an edge, then $H$ dominates $C(n)$. This conjecture is presented as a strengthening of Kalai's algebraic-shifting conjecture and would imply Turán's $(3,4)$-conjecture. No general resolution is given.