Polynomial Whitehead-minimal orbit conjecture

Let $F_r$ be a free group of rank $r$, let $F_r^k$ denote the set of $k$-tuples of elements of $F_r$, and let $U_{\min}$ be a minimal representative of the orbit of $U$ under the relevant Whitehead automorphisms. Write $\operatorname{Orb}_{\min}(U)$ for the set of minimal elements in this orbit. Polynomial Whitehead-minimal orbit conjecture. For every $U\in F_r^k$, there exists a polynomial $P_{r,k}$ such that

$$|\operatorname{Orb}_{\min}(U)|\leq P_{r,k}(|U_{\min}|).$$

This conjecture concerns the size of the set of minimal representatives and would yield a substantially better complexity bound for the Whitehead algorithm. The source gives no resolution status.