Strong Bergman asymptotics for vector-valued polynomial spaces

Let $(K,w,v,\b5mu)$ satisfy the weighted Bernstein Markov property. Let $b_1,\dots,b_N$ be any orthonormal basis of $\mathscr{P}_{r,n}U$ with respect to the product induced by $(K,w,v,\mu)$. Define the vector measures

$$T^{(r)}(\omega):=\int_K\operatorname{Re}\left(\omega,\frac{\sum_{h=1}^N\overline{(b_{h,w}(x),v)_U}b_{h,w}(x)}{N}\odot \bar v\right)_U\,d\mu(x).$$

Strong Bergman asymptotics. The measures satisfy

$$T^{(r)}\rightharpoonup^*\frac{1}{s(2\pi)^n}\sum_{l=1}^s(\mathord\cdot,u_l)_U\left(\operatorname*{dd^c}V_{K,Q_l}^*\right)^n,$$

where $Q_l=-\log w_l$ and $\left(\operatorname*{dd^c}V_{K,Q}^*\right)^n$ is the measure defined in the cited subsection. The same convergence is asserted when $(K,v,\mu)$ is replaced by any Bernstein Markov sequence $(K,v^{(r)},\mu^{(r)})$. This conjecture proposes a vector-valued analogue of strong Bergman asymptotics, identifying the weak-* limit of normalized Bergman-type vector measures; its resolution status is not established by the supplied text.