Finite Agler norm for all rational inner functions

Ascertain whether every rational inner function φ on D^d has finite Agler norm, that is, whether sup_T \|φ(T)\| < ∞ over all commuting strictly contractive d-tuples T.

Background

The Agler norm |f|_A is defined as the supremum of |f(T)| over commuting strictly contractive operator d-tuples T. In d ≥ 3, even bounded analytic functions can have infinite Agler norm. The paper notes phenomena where multiplying a rational inner function by a monomial can reduce its Agler norm to 1.

The question, suggested by V. Vinnikov and posed explicitly here, asks whether rational inner functions—potentially with boundary singularities—always have finite Agler norm, a property known to hold when the denominator polynomial has no zeros on the closed polydisk. Resolving this would deepen understanding of operator-theoretic behavior of rational inner maps in higher dimensions.