Helton–McCullough conjecture on convex positivity domains

Let $p$ be a symmetric noncommutative polynomial, and let

$$\mathcal D_p:=\bigcup_{n\geq 0}\mathcal D_p^n$$

be its positivity domain, where $\mathcal D_p^n$ is the connected component containing $0$ of

$$\mathcal P_p^n:=\{X=\{X_1,\ldots,X_g\}:X_j\in\mathbb R^{n\times n}_{sym}\text{ and }p(X)\succ0\}.$$

Here, irreducible is the condition imposed on $p$ in the conjecture, and a domain of the displayed form is called a ball when it is $\mathcal D_q$ for a polynomial

$$q(x)=c-\sum_j^w(a_j+L_j(x))^T(a_j+L_j(x)),$$

where $a_j,c$ are real numbers and $L_j(x)=b_1x_1+\cdots+b_gx_g$ is linear.

Helton–McCullough conjecture. If $p$ is irreducible and $\mathcal D_p$ is convex, then $\mathcal D_p$ is a ball.

This conjecture concerns the structure of convex positivity domains of symmetric noncommutative polynomials: convexity is expected to force the domain to have a ball-type representation. The supplied text does not state whether the conjecture has been resolved.