Degree sequences realizing labelled $h$-factors

For a positive integer $k$, let $[k] = \{1, 2, \ldots, k\}$. Let $h$ be a non-negative integer, and let $n$ be a multiple of $h + 1$. Define $H$ as the disjoint union of $n/(h+1)$ cliques (each of size $h + 1$) with vertex sets $V_1, \ldots, V_{n/(h+1)}$, where $V_i = \{ v_j \mid j = (i-1)(h+1) + k, k \in [h+1] \}$ for $i \in [n/(h+1)]$. A non-increasing integer sequence $(d_1, \ldots, d_n)$ is $H$-realizable if there exists a graph $G$ with $V(G) = V(H) = \{ v_i \mid i \in [n] \}$, $d_G(v_i) = d_i$ for all $i\in [n]$, and $G$ contains $H$ as a spanning subgraph. If $h = 0$, then a non-increasing integer sequence $(d_1, \ldots, d_n)$ is $H$-realizable if and only if there exists a graph $G$ with degree sequence $(d_1, d_2, \dots, d_n)$; Erd\H{o}s and Gallai established a necessary and sufficient condition for this property. Recently, Briggs, McDonald, and Shan extended their result to the case $h = 1$. In this paper, we establish a necessary and sufficient condition for a sequence $(d_1, d_2, \dots, d_n)$ to be $H$-realizable for any non-negative integer $h$, thereby confirming a conjecture due to Briggs, McDonald and Shan.