Mutual asymptotic Fekete sequence existence conjecture

Let $(L_1,\phi_1),\ldots,(L_m,\phi_m)$ be a collection of pairs as above, with normalized volumes as defined in the surrounding text. For each pair, let $\mu_{(L_j,\phi_j)}$ denote its associated equilibrium measure, and call a sequence of configurations a mutual asymptotic Fekete sequence if it is asymptotically Fekete for every normalized pair. Mutual asymptotic Fekete sequence conjecture. The collection admits a mutual asymptotic Fekete sequence if and only if

$$\mu_{(L_1,\phi_1)}=\cdots=\mu_{(L_m,\phi_m)}.$$

The equality of equilibrium measures is known to be necessary by equidistribution; the conjecture asserts that it is also sufficient for the simultaneous existence of asymptotically Fekete configurations.