The pre-colored leaf extension conjecture

Let $f(k)$ be the smallest integer $f$ such that for every $d\in[k]$ the following holds. Let $G$ be an arbitrary finite graph such that every degree is at most $d$, except at most one vertex of degree $d+1$. Suppose that at most $d-1$ leaves, that is, edges with one of their endpoints having degree $1$, are pre-colored. Then this pre-coloring can be extended to an edge-coloring of the whole graph $G$ that uses at most $d+f$ different colors. Pre-colored leaf extension conjecture. $f(k)=1$ for all $k\geqslant 1$. This conjecture would strengthen Vizing's theorem by asserting that the specified pre-coloring can always be extended using only $d+1$ colors, even when one vertex has degree $d+1$.