Nonfull-queue routing property for minimum loss policies

Consider the static-routing optimization problem

$$\operatorname*{\textup{minimize}}_{\boldsymbol{\pi} \in \widetilde{\boldsymbol{\Pi}}^{\textup{SR}}} \lambda {\mathsf{E}}^{\boldsymbol{\pi}}\left[\sum_{k \in \mathbb{K}} \big(1-\widetilde{A}_k\big)1_{\{\widetilde{X}_k=n_k\}}\right],$$

where $\widetilde{\boldsymbol{\Pi}}^{\textup{SR}}$ is the class of stationary randomized policies that route every arrival to some queue, possibly a full queue. Nonfull-queue routing property. An optimal policy routes an arrival to a nonfull queue, if any is available. This property would establish the equivalence between the reformulated problem allowing arrivals to be routed to full queues and the original admissible-routing problem; its status is not determined by the supplied text.