Order-of-vanishing conjecture for Gross's conjecture

With $K/F$, $G$, $V$, and $D'=\prod_{v\in V}I_{G_v,G}\subset\mathbb{Z}[G]$ as above, let $\Theta_{K,S,T}$ be the Stickelberger element. Order-of-vanishing conjecture.

$$\Theta_{K,S,T}\in D'.$$

This is the order-of-vanishing part implied by Gross's leading term conjecture, asserting that the Stickelberger element vanishes to the order forced by the places in $V$. The source gives no resolution status.