Existence conjecture for the PI-exponent of finite-dimensional algebras with involution

Let $A$ be a finite-dimensional algebra with involution $*\colon A\to A$. Its $*$-codimensions are denoted by $c_n^*(A)$, and when the limit exists its $*$-PI-exponent is

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

Existence conjecture for the $*$-PI-exponent. For any finite-dimensional algebra $A$ with involution $*$, its $*$-PI-exponent $\operatorname{exp}^*(A)$ exists.

The preceding bound shows that the sequence of $*$-codimensions is exponentially bounded, making existence of the limit the central issue. The source does not provide evidence of resolution for this conjecture.