Suppose we have a graph whose edges are coloured. It's not necessarily a proper colouring: a given node may have 0, 1, or several incident edges of a given colour.
Is the following problem NP-complete? Determine whether there are two edge-disjoint spanning trees, such that in each individual tree, no colour appears twice.
I am curious because the variant "determine whether there are two edge-disjoint spanning trees, such that in the union of the trees, no colour appears twice" is solvable in polynomial time, for example using matroid theory.