Positselski's uniform symbol-length conjecture

Let $p$) be a prime, let $\bar G$ be a pro-$p$ group, let $n$ be a positive integer, and let $\bar\omega\in H^n(\bar G,\mathbb{F}_p)$. For a field $F$ containing a root of unity of order $p$, write $G_F$ for its absolute Galois group and let $\rho\colon G_F\to\bar G$ be a profinite group homomorphism, with induced pullback $\rho^*$. Positselski's uniform symbol-length conjecture. There is a non-negative integer $M=M(\bar G,n,\bar\omega)$ such that

$$\operatorname{sl}(\rho^*(\bar\omega))\leq M.$$

The conjecture would provide uniform constructive bounds on symbol lengths of pullbacks of fixed cohomology classes, complementing the nonconstructive Norm Residue Theorem and its applications to Massey products. The supplied source gives no resolution evidence.