Piecewise regression w/ unknown breakpoint in Stan

lm_segs.stan

data {
  int<lower=1> n;      
  vector[n] x;
  vector[n] y;
}

parameters {
  real alpha;
  vector[2] beta;
  real<lower=0> sigma;
  real cutpoint;
} 

transformed parameters{
  vector[n] x2; // indicator variable for whether x_i > cutpoint
  
  for (i in 1:n) {
    if (x[i] < cutpoint) {
      x2[i] = 0;
    } else {
      x2[i] = 1;
    }
  }
}

model {
  vector[n] mu;
  alpha ~ normal(0, 1);
  beta ~ normal(0, 1);
  sigma ~ normal(0, 2);
  cutpoint ~ normal(0, 1);
  
  for(i in 1:n){
    mu[i] = alpha + beta[1] * x[i] + beta[2] * (x[i] - cutpoint) * x2[i];
  }
  y ~ normal(mu, sigma);
}

segs.R

library(rstan)
library(scales)
d <- read.csv('regression.db.csv')

stan_d <- list(y = d$direction, 
                  x = c(scale(d$date)),
                  n = nrow(d))

minit <- stan("lm_segs.stan",
              data = stan_d, 
              chains = 1, 
              iter = 2)
mfit <- stan(fit = minit , 
             iter = 1000, 
             data = stan_d, 
             chains = 3, 
             cores = 3)
mfit
post <- rstan::extract(mfit)

plot(stan_d$x, stan_d$y, 
     xlab = 'Time', ylab = 'Stuff')
for (i in seq_along(post$lp__)) {
  segments(x0 = min(stan_d$x), x1 = post$cutpoint[i], 
           y0 = post$alpha[i] + post$beta[i, 1] * min(stan_d$x), 
           y1 = post$alpha[i] + post$beta[i, 1] * post$cutpoint[i], 
           col = alpha(3, .1))
  segments(x1 = max(stan_d$x), x0 = post$cutpoint[i], 
           y1 = post$alpha[i] + 
             post$beta[i, 2] * (max(stan_d$x) - post$cutpoint[i]) + 
             post$beta[i, 1] * max(stan_d$x), 
           y0 = post$alpha[i] + post$beta[i, 1] * post$cutpoint[i], 
           col = alpha(2, .1))
}
points(stan_d$x, stan_d$y, pch = 19)

rug(post$cutpoint)