Asymptotically optimal degree balance in regular graphs

Let $G$ be a $d$-regular graph of order $n$, and let $m(H,k)$ denote the number of vertices of degree $k$ in a spanning subgraph $H$. Asymptotic degree-balance conjecture. For every $d$ there is a finite family $\mathcal{D}_d$ of exceptional graphs such that every $d$-regular graph $G$ of order $n$ not belonging to $\mathcal{D}_d$ has a spanning subgraph $H$ for which, for every integer $k$ with $0\leq k\leq d$, if $\frac{n}{d+1}$ and $\left\lceil\frac d2\right\rceil$ are odd integers, then

$$\left|m(H,k)-\frac{n}{d+1}\right|\leq 1;$$

otherwise,

$$\left|m(H,k)-\frac{n}{d+1}\right|\leq \frac{d}{d+1}$$

when $d$ is even, and

$$\left|m(H,k)-\frac{n}{d+1}\right|\leq \frac{d-1}{d+1}$$

when $d$ is odd. This conjecture refines the degree-balance problem by asserting that, apart from finitely many exceptions for each $d$, the parity obstruction is the only obstruction to attaining the bound $1$. The source explicitly notes that even the case $d=4$ is open.