Baril–Togni's multigraph neighbourhood distinguishing conjecture

Let $G$ be a connected multigraph. Write $\mu(G)$ for its edge multiplicity, $\Delta(G)$ for its maximum degree, and $\operatorname{ndi}(G)$ for the neighbourhood distinguishing index.

Baril–Togni's conjecture. If $G\neq C_5$, then

$$\operatorname{ndi}(G)\leq \Delta(G)+\mu(G)+1.$$

This extends the neighbourhood distinguishing index conjecture from simple graphs to multigraphs. The source presents it as an open strengthening.