The inclusion chromatic index bound for connected graphs

Let $G$ be a connected graph with minimum degree $\delta\geq 2$ and maximum degree $\Delta$, and let $\chi'_{\subset}(G)$ denote its inclusion chromatic index, the least number of colours in a proper edge colouring such that the palette at every vertex is not contained in the palette at any neighbour. Inclusion chromatic index conjecture. If $G$ is not isomorphic to $C_5$, then

$$\chi'_{\subset}(G) \leq \left\lceil\left(1+\frac{1}{\delta-1}\right)\Delta\right\rceil.$$

The conjectured bound matches the lower bound supplied by the paper's infinite family of examples and would therefore be sharp. The paper proves an upper bound with an additive constant for every fixed $\delta\geq 2$, with $\frac{1}{\delta-1}$ replaced by $\frac{4}{\delta-1}$; the stated bound remains open.