A negative-integer integrality conjecture for equivariant L-values

Let $L/K$ be a Galois extension of number fields with Galois group $G$, let $r<0$, and let $\mu_{1-r}(L):=(\mathbb{Q}/\mathbb{Z})(1-r)^{G_L}$, where $G_L$ is the absolute Galois group of $L$. For a finite set $S$ of places containing all ramified and infinite places, let $\theta_S(r)$ be the corresponding equivariant $L$-value. Negative-integer integrality conjecture. For every $x\in\operatorname{Ann}_{\mathbb{Z}G}(\mu_{1-r}(L))$, one has

$$\operatorname{nr}(x)\cdot\theta_S(r)\in\mathcal I(G)$$

for all such finite sets $S$. This is presented as a partial analogue of the integrality conjecture at $r=0$; the source explicitly says that no general conjecture in complete analogy is asserted, because no convincing reason is known for it in general.