Monotonicity conjecture for the number of rainbow trees

Let $k,c,c'\in\mathbb{N}$ with $c'\leq c$, and let $G$ be a graph. For $r\in\{c,c'\}$, let $T(r,k)$ be the random variable counting the number of rainbow trees of order $k$ when the edges of $G$ are coloured uniformly with $r$ colours. Rainbow-tree monotonicity conjecture. The random variable $T(c,k)$ stochastically dominates $T(c',k)$.

A coupling establishing this monotonicity is immediate when $c=2c'$, but a general coupling was not found. The conjecture formalises the intuition that using more colours should make rainbow substructures more abundant.