Formanek's minimal-degree conjecture for central polynomials

Let $K$ be a field of characteristic $0$, and let $M_d(K)$ denote the algebra of $d\times d$ matrices. A central polynomial for $M_d(K)$ is a polynomial that takes central values on $M_d(K)$ without being a polynomial identity. Formanek's minimal-degree conjecture. The minimal degree of the central polynomials for $M_d(K)$ is equal to

$$\frac{1}{2}(d^2+3d-2).$$

This conjecture refines the earlier $d^2$ expectation; the supplied text notes that the minimum is $8$ for $d=3$, which agrees with the displayed formula, but gives no general resolution.