Capacity lower-bound conjecture for compact non-pluripolar sets

Let $A=B_c(0,1)$ and $Omega=B_c(0,a)$. For a compact non-pluripolar set $E\bigsubset A$, let $u_{E, Omega}$ denote the relative extremal function and $V_E$ the Siciak extremal function.

Capacity lower-bound conjecture. There exists a constant $C_{a,n}>0$ such that, for every compact non-pluripolar set $E\bigsubset A$,

$$\left|\sup_A u_{E,\Omega}\right|\sup_{\Omega}V_E\geq C_{a,n}.$$

This conjecture would extend the Bernstein doubling inequality and the associated estimate from Lebesgue measure to Siciak capacity, allowing analogous results when $E$ does not have positive Lebesgue measure. Its resolution status is not specified in the source.