Polynomial Whitehead-descent time-complexity conjecture

Let $F_r$ be a free group, let $NMin\subset F_r$ be the set of non-minimal elements, and let $WC(w)$ denote Whitehead complexity. Polynomial Whitehead-descent time-complexity conjecture. The time complexity, or at least the average-case time complexity, of Problem A on inputs $w\in NMin$ is bounded above by

$$P(r)WC(w)|w|,$$

where $P(r)$ is a fixed polynomial. This conjecture seeks a polynomial-in-rank complexity bound for solving Problem A on non-minimal inputs, proportional to Whitehead complexity and word length. The source gives no resolution status.