Whether R-hat=NA is a problem or not depends on which parameters are causing it. The upper halves of those Cholesky factors are by definition zero so their R-hat is undefined. It’s safe to ignore them.
Look at the summary and see if any non-fixed parameters with high R-hat.
That should be
f_sd[t] ~ multi_normal_cholesky(f_sd[t-1],
diag_pre_multiply(tau, sigma_f_sd));
The covariance matrix diagonal is variances, not standard deviations.
Q[t] = tcrossprod(diag_post_multiply(beta[t], f_sd[t]))
+ diag_matrix(square(y_sd));
Cauchy distribution has a long tail. The prior expects that one tau will be much larger than the others.
A better prior would be tau ~ normal(0, max_tau); where max_tau is the largest value tau could reasonably be.