Asymptotic empirical-current conjecture for Fekete currents

Let $E\subset\mathbb{R}^n$ be a compact, non-pluripolar set, and let $\\{\mathcal F^{(r)}\\}=\\{\mathcal F^{1,r},\dots,\mathcal F^{N(r),r}\\}$ be an asymptotic sequence of currents with $\mathcal F^{s,r}\in\mathscr{A}^k(E)$ for $s=1,\dots,N(r)$. Define its empirical current by

$$T^{(r)}:=\frac{1}{N(r)}\sum_{s=1}^{N(r)}\mathcal F^{s,r}.$$

Asymptotic empirical-current conjecture. The total variation measures $\lVert T^{(r)}\rVert$ converge to $\mu_E$ in the weak$^*$ topology of measures supported on $E$. Moreover, if $T\in\mathscr{S}_0^k(E)$ is an accumulation point of $\\{T^{(r)}\\}$ in $\mathscr{D}_{0,k}(E)$, then there is an orthonormal basis $\\{\tau_1,\dots,\tau_n\\}$ of $\mathbb{R}^n$ such that

$$T(\omega)=\frac{1}{\binom{n}{k}}\sum_{|\alpha|=k}'\int \langle\omega(x);\tau_{\alpha_1}\wedge\dots\wedge\tau_{\alpha_k}\rangle\\,\mathrm{d}\mu_E.$$

The conjecture seeks the analogue for asymptotically Fekete currents of the known equidistribution of true and approximate Fekete points. The paper explains that existing results for Fekete currents require a restrictive form of the currents and therefore do not establish this stronger asymptotic statement; its resolution remains open.