Has "deep vs. wide" been resolved? All else being equal, including total neuron count, I give the following definitions:

wide is a parallel ensemble, where good chunks of the neurons have the same inputs because the inputs are shared and they have different outputs.
deep is a series ensemble, where for the most part neurons have as input the output of other neurons and few inputs are shared.

For CART ensembles the parallel (wide) ensemble is a random forest while the series (deep) ensemble is a gradient boosted machine.  For several years the GBM was the &quot;winningest&quot; on kaggle.
Is there a parallel of that applied to Neural networks?  Is there some reasonable measure that indicates whether deep outperforms wide when it comes to neural networks?  If I had the same count of weights to throw at a tough problem, all else being equal should they be applied more strongly in parallel or in series?
I am not sure what you are really looking for but I leave here this paper here, where some intuition into that direction is provided. This paper compares the performance of a deep learning model scaling in 3 dimensions: resolution, width and depth. As depicted in their definition:

If you go to section 3.2 you will see how scaling the different dimensions independently (resolution, width and depth) affects performance and how by performing a compound scaling the performance of the model is maximized (so there is a close relation). It is a very thorough ablation study. For me this was the paper where I understood how these Width, Depth and Resolution parameters come together.
Rethinking Model Scaling for Convolutional Neural Network: https://arxiv.org/abs/1905.11946