Degree sequences realizing labelled $h$-factors

For a positive integer $k$, let $[k]=\{1,2,\dots ,k\}$. Let $h$ be a non-negative integer, and let $n$ be a multiple of $h+1$. Define $H$ as the disjoint union of $n\mathrm{/}(h+1)$ cliques (each of size $h+1$) with vertex sets $V_1,\dots ,V_{n\mathrm{/}(h+1)}$, where $V_i=\{v_j\mid j=(i-1)(h+1)+k,k\in [h+1]\}$ for $i\in [n\mathrm{/}(h+1)]$. A non-increasing integer sequence $(d_1,\dots ,d_n)$ 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 $(d_1,\dots ,d_n)$ is $H$-realizable if and only if there exists a graph $G$ with degree sequence $(d_1,d_2,\dots ,d_n)$; Erdős 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.