Balogh-Kostochka-Treglown degree-sequence conjecture for equitable colorings

Let $n,k∈\mathbb{N}$ with $k≤n$. Let $G$ be an $n$-vertex graph whose degree sequence is $d_1≥⋯≥d_n$. Let an equitable $k$-coloring be a proper $k$-coloring whose color classes have sizes differing by at most one. Degree-sequence equitable-coloring conjecture. If

$$d_i<2k-i\quad\text{for all }i<k,$$

and

$$d_{k+1}<k,$$

then $G$ contains an equitable $k$-coloring.

This conjecture is proposed as a degree-sequence analogue of the Hajnal-Szemerédi conjecture, inspired by work of Balogh, Kostochka, and Treglown. Its general status is open.