Linear-query priority-queue algorithm for maximum st-flow in directed planar graphs

Let $G$ be a directed planar graph with $n$ vertices and source $s$ and sink $t$. Consider an augmenting-path algorithm for computing a maximum $st$-flow in $G$.

Priority-queue implementation conjecture. Such an algorithm can be implemented with $O(n)$ queries to a priority queue.

This proposes extending the priority-queue implementation known for the $st$-planar case to directed planar graphs without requiring the source and sink to lie on a common face. The paper presents it as a promising direction for developing practical maximum-flow algorithms; its status is not resolved in the supplied text.