Do multicolour Ramsey numbers of trees grow linearly: is $R_k(T)\leq kn+O(1)$? (Erdős #557)

For a graph $G$, let $R_k(G)$ denote the $k$-colour Ramsey number: the least $m$ such that every colouring of the edges of $K_m$ with $k$ colours contains a monochromatic copy of $G$. Is it true that, for each fixed number of colours $k$, $$R_k(T)\leq kn+O(1)$$ for every tree $T$ on $n$ vertices, where the implied additive constant is uniform over all trees on $n$ vertices (in particular, it does not grow with $n$)?

Acceptance. FULLY RESOLVES: a complete proof — machine-checkable (Lean/Coq) preferred, else a full written proof — that for each fixed $k$ there is a constant $c_k$ with $R_k(T)\leq kn+c_k$ for every tree $T$ on $n$ vertices; OR a counterexample, i.e. a family of trees $T$ (with $|T|=n\to\infty$) and colourings witnessing $R_k(T)-kn\to\infty$. ADVANCES: prove the bound for a restricted infinite tree family (paths, brooms, spiders, or bounded-degree trees), or for a fixed small number of colours $k$, or conditionally on the Erdős–Sós conjecture (clearly flagged), or prove the weaker bound $R_k(T)\leq kn+o(n)$ — each with proof; OR compute $R_k(T)$ exactly for all trees on $n\leq N$ vertices and small $k$ with exhaustiveness certificates, reporting the attained $(N,k)$ and the observed additive gap. Deliver the proof file, the counterexample plus certificate, or the verification code plus certified table.