Conjecture on edge-disjoint 1-factors in realizations of degree sequences

Let $\mathrm{\pi}=(d_{1},\ldots,d_{n})$ be a degree sequence with $n$ even. A $1$-factor is a spanning subgraph in which every vertex has degree $1$, and edge-disjoint $1$-factors share no edges.

Edge-disjoint 1-factor conjecture. Some realization of $\mathrm{\pi}$ has $k$ edge-disjoint $1$-factors if and only if

$$(d_{1}-k,\ldots,d_{n}-k)$$

is graphic.

This would strengthen Kundu's $k$-factor theorem and generalize known results on realizations containing multiple edge-disjoint $1$-factors. The source presents it as a conjecture and gives only a partial result, so its general status remains open.