Or the other way around, an example of a family of three commuting contractions for which there exists an isometric dilation, but which does not satisfy the von Neumann inequality?
The only example I've come across is Parrot's example in "Unitary dilations for commuting contractions", where you take:
$$T_1 = \begin{pmatrix}0&0\\ I&0\end{pmatrix}$$
$$T_2 = \begin{pmatrix}0&0\\ U&0\end{pmatrix}$$
$$T_3 = \begin{pmatrix}0&0\\ V&0\end{pmatrix}$$
where U and V are two non commuting unitary operators. Here, the family $T_1, T_2, T_3$ does not have an isometric dilation, but it satisfies the von Neumann inequality.
Nevertheless, it turned out to be not that easy to prove this, so my question is, is there any other such example which is easier to prove?