Realization conjecture for PI-exponents of algebras with involution

Let $\alpha\geq 1$ be a real number. An algebra with involution $A_\alpha$ has a $*$-PI-exponent, when the limit exists, defined by

$$\operatorname{exp}^*(A_\alpha)=\lim_{n\to\infty}\sqrt[n]{c_n^*(A_\alpha)}.$$

Realization conjecture for $*$-PI-exponents. For any real value $\alpha\geq 1$, there exists an algebra $A_\alpha$ with involution such that its $*$-PI-exponent exists and

$$\operatorname{exp}^*(A_\alpha)=\alpha.$$

This conjecture describes the proposed range of values of the $*$-PI-exponent. The source motivates it using results cited as GMZ, but gives no resolution in the supplied text.