Horseshoe prior can be hard to fit see Sparsity information and regularization in the horseshoe and other shrinkage priors. Regularised_horseshoe have been proposed as better alternative Bayes Sparse Regression - reg. horseshoe for multinomial model - #7 by stemangiola (also harder to fit than normal distribution)
If you assume sparsity over your parameter, N(0,1) will not model it and will lead to poor estimates of the ~ 0 parameters and shrunk estimated for the far-from-0 parameters. See Bayes Sparse Regression
@avehtari feel free to correct me in case you think is appropriate.