Degree-sequence conditions for r-Hamiltonian hypergraphs

Let $(d_1,d_2,\ldots,d_n)$ be an integer sequence with

$$d_1\leq d_2\leq \dots\leq d_n,$$

where $n>2r$ and $r\geq 3$. A hypergraph is $r$-Hamiltonian if it remains Hamiltonian after the deletion of any set of fewer than $r$ vertices. Degree-sequence conjecture. If

$$d_i>i \qquad\text{for }1\leq i<r,$$

if

$$d_i\leq \binom{i}{r-1}\quad\text{then}\quad d_{n-i}>\binom{n-i-1}{r-1}\qquad\text{for }r\leq i\leq\left\lfloor\frac{n-1}{2}\right\rfloor,$$

and, when $n$ is even, if

$$d_{\frac{n-2}{2}}\leq \binom{\frac{n-2}{2}}{r-1}+1,$$

then

$$d_{\frac{n+2}{2}}>\binom{\frac{n}{2}-2}{r-1}+\left(\frac{n}{2}+1\right)\binom{\frac{n}{2}-2}{r-2},$$

then the sequence is $r$-Hamiltonian. This proposes degree-sequence conditions that would generalize Chvátal's theorem to hypergraphs; the authors expect Chvátal's condition to be sufficient but not necessary in this setting.