Maximal dimension conjecture for Kummer spaces in tensor products of cyclic algebras

Let $d$ be a positive integer, let $F$ be an infinite field of characteristic $0$ containing a primitive $d$th root of unity, and let

$$A=\bigotimes_{k=1}^n \left(\alpha_k,\beta_k\right)_{d,F}$$

be a division algebra, where each factor is a cyclic algebra of degree $d$. An $F$-vector subspace of $A$ is called a Kummer space if every element $v$ in it satisfies $v^d\in F$. Maximal dimension conjecture. The maximal dimension of a Kummer space in $A$ is $nd+1$. The conjecture would determine the largest Kummer spaces available for applications to symbol-length bounds. The paper's abstract proves the corresponding value $4n+1$ for the generic tensor product of $n$ cyclic algebras of degree $4$, but the stated claim for an arbitrary division algebra and positive integer $d$ is not resolved here.