Almost equitable factorization conjecture with controlled parity defects

Let $k$ be a positive integer, let $\varepsilon_1,\ldots,\varepsilon_k$ be nonnegative real numbers satisfying

$$\varepsilon_1+\cdots+\varepsilon_k=1,$$

and let $G$ be a graph. An edge-decomposition into factors is a decomposition of $E(G)$ into the edge sets of factors $G_1,\ldots,G_k$. For each vertex $v$, let $j_v$ denote an index whose factor has odd degree at $v$, when such an index exists.

Almost equitable factorization conjecture. The graph $G$ can be edge-decomposed into factors $G_1,\ldots,G_k$ such that, for every $v\in V(G)$,

$$\left|d_{G_i}(v)-\varepsilon_i d_G(v)\right|<2\qquad(1\le i\le k),$$

and at most one factor has odd degree at $v$. In particular, no such index exists when $d_G(v)$ is even. Furthermore, if $G$ is odd-$\lceil 1/\varepsilon_1\rceil$-edge-connected, then $j_v=1$ for every odd-degree vertex $v$.

The source says that the conjecture is proved when almost all proportions are equal, except possibly $\varepsilon_1$, and that a weaker version with upper bound $6$ is confirmed. The full statement remains open.