Half-degree conjecture for complementary tuples in balanced multipartite hypergraphs

Let $H$) be an $n$-balanced $r$-partite $r$-graph, and let $I$ be a subset of $[r]:=\{1,2,\ldots,r\}$. An $I$-tuple is an element of $\times_{i\in I}V_i$, and write $I^c:=[r]\setminus I$. For an $I$-tuple $f$ and an $I^c$-tuple $g$, let $d(f)$ and $d(g)$ denote their degrees in $H$. Half-degree conjecture. If

$$d(f)>\frac{n^{r-|I|}}{2}$$

for every $I$-tuple $f$ and

$$d(g)\geq\frac{n^{|I|}}{2}$$

for every $I^c$-tuple $g$, then $H$ has a perfect matching. Thus, each tuple has degree at least roughly half of its possible degree, with a strict inequality on the $I$-side. The condition would weaken the sharp minimum-degree hypothesis proved earlier in the paper; its general validity is presented as an open problem.