Are non-crossing ROC curves sufficient to rank classifiers by expected loss? We have two models outputting estimates of class probabilities. Combined with a probability cutoff / threshold, these yield classification decisions: if the estimated probability of class 1 is above the threshold, the assigned label is class 1; otherwise, it is class 0. We want to compare the models in terms of their estimated expected loss for a given threshold. The loss function $L(\hat{Y},Y)$ is given by
\begin{aligned}
L(0,0)&=0, \\
L(0,1)&=a, \\
L(1,0)&=b, \\
L(1,1)&=0
\end{aligned}
with $a,b>0$. The estimated losses on a test subsample are not available. However, we have the ROC curves on the test subsample for each model. We do not have the data behind the two ROC curves, but we observe visually that the first ROC curve entirely dominates the second one, i.e. the two curves never cross (only touch in the bottom left and top right corners).
Question: Is that sufficient to conclude that the first model has lower estimated expected loss than the second one (for a given threshold)? If not, could you offer a counterexample?
 There was a bounty put on this question, but I have not received a satisfactory answer. If you are able to answer the question, I will consider awarding another bounty to it. 
 Related question: "Is a pair of threshold-specific points on two ROC curves sufficient to rank classifiers by expected loss?". 

ROC curves are incompatible with optimal decision making because each point on the curve conditions on the future to predict the past.  See https://www.fharrell.com/post/mlconfusion.  For useful measures of predictive performance see https://fharrell.com/post/addvalue.
Expected loss is computed from measures that respect the direction of information flow, e.g., predicting the future from the past.
