NUTS basically can’t handle the constraints – even with tuning the accept rate way up (0.99) and drastically increasing the tree depth (25) it is a very unhappy sampler. Possibly Gibbs Sampling would be better since it would give more granular control over the integration limits (in particular the Gibbs pass over the ordered diagonal).
It’s also possible that the hard limit at 0 is causing a significant problem, so a latent formulation for a lognormal problem may help ADVI (i.e. W_lat ~ N(), W_d = exp(W_lat)). Or an auxiliary exponential formulation with Y1, ..., Yn ~ exp(b), X1 = Y1, X2 = Y1 + Y2, … But these really constrain the space of priors over the diagonal.