Perfectness conjecture for preorderings of matrix polynomials

Let $cmathcal{B}=cmathbb{R}[x_1,\cdots,x_n]$, let $S=cbrace p_1,\cdots,p_mcbracecsubsetcmathcal{B}$, and let $T_S$ be the preordering generated by $S$. Set $cmathcal{A}=M_N(cmathcal{B})$ and let $cmathfrak{tr}colon M_N(cmathcal{B})ctocmathcal{B}$ be the normalized trace.

Perfectness conjecture. The induced quadratic module satisfies

$$cmathop{\operatorname{Ind}}^{cmathfrak{tr}} T_S=T_SccdotcSigma M_N(cmathcal{B})^2,$$

i.e. $T_S$ is a perfect quadratic module.

If $T_S$ is saturated, this is equivalent to representing every matrix polynomial positive semidefinite on $K_S$ as $csum_icalpha_i A_i^*A_i$ with $A_icin M_N(cmathcal{B})$ and $calpha_icin T_S$. The representation is known in several one-variable and a particular two-variable case, but the conjecture remains open in general.