Rigidity conjecture for essentially disjoint plurisupported currents

Let $T$ and $S$ be essentially disjoint plurisupported currents, meaning that they satisfy Definition, and suppose that

$$\mathrm{Supp}(T)=\mathrm{Supp}(S).$$

Rigidity conjecture. Either $T=0$ or $S=0$.

This asks whether plurisupported currents with the same support exhibit a rigidity analogous to that of currents of integration arising from rigid underlying divisors. The supplied text presents it as an interesting follow-up question, and gives no resolution.