The positivity conjecture for semi-simple TERP-structures

Let $(w,u_1,\ldots,u_n,\xi,T)$ be data with $w\in{\mathds Z}$, $u_i\in{\mathds C}$ pairwise distinct, $\xi\in S^1$ satisfying $\Re((u_i-u_j)/\xi)<0$ for $i<j$, and $T\in M(n\times n,{\mathds R})$ upper triangular with $T_{ii}=1$. These data determine a unique semi-simple mixed TERP-structure of weight $w$. Positivity conjecture. If $T+T^{tr}$ is positive definite, then the corresponding TERP-structure is pure and polarized. This proposes a partial criterion for purity and polarization in the semi-simple case; the source presents it as open.