The GDN critical exponent and the index of primitivity

Let $CE_n$ denote the critical exponent for $n$-by-$n$ generalized doubly nonnegative matrices, and let $MIP_n$ be the maximum index of primitivity among primitive $n$-by-$n$ generalized doubly nonnegative matrices. GDN critical exponent conjecture.

$$MIP_n-1<CE_n.$$

The conjecture concerns the relationship between the largest index of primitivity and the critical exponent for generalized doubly nonnegative matrices. The preceding discussion motivates it by arguing that equality or the reverse inequality would require an unlikely multiple root in an associated exponential polynomial; no resolution is given in the source.