Linear recurrence characterization of max-linearly recurrent sequences

Determine when a max-linearly recurrent sequence is also linearly recurrent, beyond the sufficient condition that one affine-linear expression in its recurrence is eventually strictly larger than all the others.

Background

The paper introduces max-linearly recurrent sequences as sequences whose terms are the maximum of finitely many linear combinations of preceding terms. Every linearly recurrent sequence is automatically max-linearly recurrent, but the converse need not be immediate because different expressions in the maximum may attain the maximum infinitely often.

The paper notes that eventual strict dominance of a single expression is sufficient to imply linear recurrence, while presenting the two-term Gale–Robinson-type recurrence in equation (\ref{mgeq4}) as a non-trivial example where both expressions realize the maximum infinitely often and yet, for specified initial values, the sequence is still linearly recurrent. A general characterization remains unresolved.