ra.rings and algebras – Are the trace relations among matrices generated by cyclic permutations?

Let $X_1,dots,X_n$ be non commutative variables such that $operatorname{tr} f(X_1,dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r geq 1$. Does this imply that $f$ is in the ideal generated by cyclic permutations: $g_1dots g_k – g_2dots g_k g_1$ for any polynomials $g_i$ in the $X_i$ and $k geq 2$?

(And if I have missed any obvious relations, is the statement true up to adding in those relations to the ideal?)