Asymptotic growth conjecture for central-polynomial codimensions

Let $A$ be a PI algebra, and let $\delta_n(A)$ denote the $n$th codimension sequence associated with its central polynomials. For ordinary codimensions, Giambruno and Zaicev proved an asymptotic of the form

$$c_n(A)\sim\alpha\,n^g d^n,$$

where $d$ is an integer, and Berele proved that $g\in\frac12\mathbb Z$. Asymptotic growth conjecture. The analogous asymptotic theorems should hold for $\delta_n(M_k(F))$, and probably for other algebras $A$. Thus, for $\delta_n(M_k(F))$, the corresponding polynomial exponent should be a half-integer and the exponential base should be an integer, although the source phrases the extension to other algebras tentatively. These asymptotics are presented as conjectural in the source.