Constant population size vs. Bayesian skyline

[Nokikana]

Dear BEAST community,

I'm constructing a dated phylogeny of whole genome sequenced bacteria with known sampling dates and would also like to construct a BSP. I have tested the strict clock and the log normal relaxed clock with combinations of different tree models; coalescent constant, exponential and Bayesian skyline.

Constant population adn Bayesian skyline tree models converge and mixing looks good (exponential fails), but as far as I understand the constant population tree model works the best, but if I use the coalescent Bayesian skyline and construct a skyline plot I see a clear increase in population size quite close to the present. Would this rule out the use the constant population tree model or at least it being the best model? Should I just use the coalescent Bayesian skyline tree model in the analyses?

Basically this might come to the fact that I don't understand very well what's the correlation of the coalescent constant population and the Bayesing skyline. Would anyone be able to explain this to me in not too complicated terms.

thanks,

N