Special labelled h-factor realization conjecture

Let $n\in\mathbb{N}$, let $d_1\ge d_2\ge\dots\ge d_n\ge1$ be integers, and let $h$ be a positive integer. The labelled h-factor is

$$\{(1,2,\ldots,h+1),\ldots,(n-h,\ldots,n)\},$$

where each tuple denotes the complete graph on its listed vertices. Special labelled h-factor conjecture. The sequence $(d_1,d_2,\ldots,d_n)$ can realize this h-factor if and only if $\sum_{i=1}^n d_i$ is even, $n$ is a multiple of $h+1$, and, for every $k\in[n]$,

$$\sum_{i=1}^k d_i\le k(k-1)+\sum_{i=k+1}^{k+1+h-s}\min\{d_i-h+s,k\}+\sum_{i=k+1+h-s+1}^n\min\{d_i-h,k\},$$

where $s\in\{0,\ldots,h\}$ and $s\equiv k\pmod{h+1}$. The conjecture is open in general; the source cites partial results and uses it as an assumption for a subsequent corollary.