A family of $2\times 2$ tetrablock contractions with non-commuting fundamental operators

In this note, we provide a family of $2\times 2$ tetrablock contractions that have tetrablock isometric dilation, but the corresponding fundamental operators do not commute. This answers a question raised by Bhattacharyya [Indiana Univ. Math. J. 2014] about the necessity of commuting fundamental operators for a tetrablock contraction to have rational dilation in a negative direction.