Halfpap's rainbow Turán conjecture for paths

Prove that the rainbow Turán number f(d), defined as the largest integer such that every properly edge-coloured graph of average degree at least d contains a rainbow path of length f(d), equals ⌈d⌉−1 up to at most an additive constant for all d>2.

Background

The paper defines f(d) as the extremal guaranteed length of a rainbow path in a properly edge-coloured graph with average degree at least d. It records lower and upper bounds that do not yet coincide.

The upper bound ⌈d⌉−1 arises from a coloured Cayley graph over an elementary abelian 2-group. The paper reports that Halfpap conjectured this upper bound to be tight for all d, possibly subject to an additive constant.