On a list $(k,l)$-coloring of incidentors in multigraphs of even degree for some values of $k$ and $l$

The problem of a list $(k,l)$-coloring of incidentors of a directed multigraph without loops is studied in the case where the lists of admissible colors for incidentors of each arc are integer intervals. According to a known conjecture, if the lengths of these interval are at least $2\Delta+2k-l-1$ for every arc, where $\Delta$ is the maximum degree of the multigraph, then there exists a list $(k,l)$-coloring of incidentors. We prove this conjecture for multigraphs of even maximum degree $\Delta$ with the following parameters:
$\bullet \ l\ge k+\Delta/2$;
$\bullet \ l< k+\Delta/2$ and $k$ or $l$ is odd;
$\bullet \ l< k+\Delta/2$ and $k=0$ or $l-k=2$.