hyper-util: client connection pool idle timeout does not work as expected

Version
Latest hyper (1.3.1) & hyper-util (0.1.3)

Platform
Windows11

Description
The issue reproduces in a modified hyper-util client example:

use std::env;

use http_body_util::Empty;
use hyper::Request;
use hyper_util::client::legacy::{connect::HttpConnector, Client};

#[tokio::main(flavor = "current_thread")]
async fn main() -> Result<(), Box<dyn std::error::Error>> {
    let url = match env::args().nth(1) {
        Some(url) => url,
        None => {
            eprintln!("Usage: client <url>");
            return Ok(());
        }
    };

    // HTTPS requires picking a TLS implementation, so give a better
    // warning if the user tries to request an 'https' URL.
    let url = url.parse::<hyper::Uri>()?;
    if url.scheme_str() != Some("http") {
        eprintln!("This example only works with 'http' URLs.");
        return Ok(());
    }

    let client = Client::builder(hyper_util::rt::TokioExecutor::new())
        .pool_timer(hyper_util::rt::TokioTimer::new())
        .pool_idle_timeout(std::time::Duration::from_secs(1))
        .build(HttpConnector::new());

    let req = Request::builder()
        .uri(url)
        .body(Empty::<bytes::Bytes>::new())?;

    for _ in 0..4 {
        eprintln!("Running request");
        let resp = client.request(req.clone()).await?;

        eprintln!("{:?} {:?}", resp.version(), resp.status());
        eprintln!("{:#?}", resp.headers());

        eprintln!("Waiting 5 seconds");
        _ = tokio::time::sleep(std::time::Duration::from_secs(5)).await;
    }

    eprintln!("Exiting");

    Ok(())
}

Client is configured with idle timeout of 1 sec and the example performs several requests with 5 second delay.
The expectation is that a connection should be dropped from the pool after 1 second during each loop iteration. In practice, the connections start to get dropped only on the second iteration of the loop.

In a real application, this results in connections lingering until a new request is issued.

Start with the modified hyper-util client example and the Client builder calls for pool_timer and pool_idle_timeout. Reproduce the four requests with a five-second delay on Windows 11, then inspect the connection-pool timeout path. Done means the idle connection is dropped after the configured one-second timeout during each loop iteration, with regression coverage for the observed behavior.