The von Neumann inequality for $3\times 3$ matrices in the unit Euclidean ball

It is shown that the constant $c_{d,3}$ in von Neumann’s inequality for d-tuples of commutative and row contractive $3\times 3$ matrices, as proved by Hartz, Richter, and Shalit in [2], is independent of the size of the d-tuple. A numerical estimation of the constant is provided.