Uniqueness of the root of the degree-eight polynomial

Determine whether the polynomial g(z) associated with the lower-bound function for the weighted shift operator T(1,sq,q^2,tq^3,q^4,sq^5,q^6,tq^7,\ldots) has a unique root in the interval (1/3,1) for every s,t>0 and 0<q<1.

Background

The polynomial g(z) is obtained from the derivative of the function used to calculate the lower bound for the numerical radius of T(1,sq,q2,tq3,q4,sq5,q6,tq7,\ldots). The paper proves that g(z) has no root in (0,1/3) and at least one root in (1/3,1).

Uniqueness is established only when s,t belong to (q,1/q). For the remaining parameter regimes, namely when at least one of s or t lies outside [q,1/q], the authors state that uniqueness remains unresolved and formulate the question explicitly.

References

For the remaining cases, i.e., when at least one of $s,t\notin [q,\frac{1}{q}]$, this question remains unresolved. Motivated by extensive examples, the following question naturally arises. For $s,t>0$ and $0<q<1$ does the polynomial $g(z)$ defined in g(z) have a unique root in $(\frac{1}{3},1)$?

— On the numerical radius of a class of weighted shift operators  (2608.17486 - Ghosh et al., 18 Aug 2026) in Section 2, immediately before the stated Open question