The asymptotic fractional matching threshold conjecture for uniform hypergraphs

Let $f_d(k,n)$ denote the minimum $d$-degree forcing a fractional matching of the relevant size in a $k$-uniform hypergraph, with $1\le d\le k-1$. Fractional matching threshold conjecture. For all $1\le d\le k-1$,

$$f_d(k,n)\sim \left\{1-\left(\frac{k-1}k\right)^{k-d}\right\}\binom{n-d}{k-d}.$$

The lower bound comes from the construction $H_1(\lceil n/k\rceil)$; the claim is confirmed asymptotically when $1\le k-d\le4$, but is not established for all parameters.