Rails Puma running out of Redis connections

I've looked around at other similar questions on SO but can't quite piece things together well enough. I have a Rails app (on Heroku) that uses Puma with both multiple processes and multiple threads. My app also uses Redis as a secondary data store (in addition to a SQL database), querying Redis directly (well, through the connection_pool gem). Here's my Puma config file:

workers Integer(ENV["WEB_CONCURRENCY"] || 4)
threads_count = Integer(ENV["MAX_THREADS"] || 5)
threads threads_count, threads_count

preload_app!

rackup DefaultRackup
port ENV["PORT"] || 3000
environment ENV["RACK_ENV"] || "development"

on_worker_boot do
  # Worker specific setup for Rails 4.1+
  ActiveRecord::Base.establish_connection

  redis_connections_per_process = Integer(ENV["REDIS_CONNS_PER_PROCESS"] || 5)
  $redis = ConnectionPool.new(size: redis_connections_per_process) do
    Redis.new(url: ENV["REDIS_URL"] || "redis://localhost:6379/0")
  end
end

My Redis instance has a connection limit of 20, and I find myself regularly going over this limit, despite having what should be (as far as I can tell) only 5 connections per process spread across 4 worker processes.

In fact, I even get max number of clients reached Redis errors when I set REDIS_CONNS_PER_PROCESS to 1. Is on_worker_boot called for each thread rather than each process?

I've also tried having a separate redis.rb initializer, which works but only when REDIS_CONNS_PER_PROCESS is 1, which seems odd since I should be able to have it up to 4 if I'm doing my math correctly (4 worker processes + 1 master process) * 4 connections per process. (Note that for the purposes of this question I'm ignoring errors that occur around deploying, since I'm assuming Heroku might be connecting both old and new processes during that process, even though I'm not using Preboot.)

Where am I misunderstanding how this all fits together?