Conjecture on valid orientations with four exceptional boundary vertices

Let $G$ be a plane graph with a valid prescription function $p:V(G)\rightarrow\{-1,0,1\}$, a specified face $F_G$, and at most four specified vertices $d,r,s,t$. The vertex $d$, when present, is a degree-$3$ boundary vertex that may be oriented; each of $r,s,t$, when present, is a degree-$3$ boundary vertex. Assume

$$G\text{ is 3-edge-connected,}$$

$G$ has at most four $3$-edge-cuts, only $\delta(\{d\})$, $\delta(\{r\})$, $\delta(\{s\})$, or $\delta(\{t\})$, and every vertex outside the boundary of $F_G$ has five edge-disjoint paths to that boundary.

Conjecture on valid orientations with four exceptional boundary vertices. Then $G$ has a valid orientation.

This conjecture extends the paper's orientation theorem by allowing further degree-$3$ boundary vertices and is presented as best possible in view of related counterexamples.