Generation conjecture for tensor products of cyclic algebras

Let $F$ be a field, let $d$ be a positive integer, and let

$$A=\bigotimes_{k=1}^n(\alpha_k,\beta_k)_{d,F}=\bigotimes_{k=1}^n F[x_k,y_k]$$

be a tensor product of cyclic algebras. Define

$$X=\sum_{k=1}^n x_1^{-1}\dots x_{k-1}^{-1}x_k y_1\dots y_{k-1},\qquad Y=\sum_{k=1}^n x_1^{-1}\dots x_{k-1}^{-1}y_1\dots y_k.$$

A subspace is $d$-central when every element has its $d$th power in $F$ and its lower positive powers are not central.

Generation conjecture. The algebra $A$ is generated by $X$ and $Y$.

The elements $X$ and $Y$ span a $d$-central subspace, so the conjecture would produce a finite linearization of the exponentiation form. Together with the rank monotonicity conjecture and the stated theorem, this would yield finite linearizations of unbounded high ranks for forms of degree at least four in at least two variables. The supplied text gives no proof or counterexample.