Invalid Signature while Connecting to Instapaper's Xauth on PHP

Question

I've been trying to get a PHP application to connect to Instapaper's Xauth services, but for the life of me I can't get it to work. I keep getting an "403: Invalid signature." error.

The error says that my signature base string wasn't what it expected, but when I compare the signature base string I construct with what they say they expect, they're exactly the same (sensitive information removed):
My signature base string:

POST&https%3A%2F%2Fwww.instapaper.com%2Fapi%2F1%2Foauth%2Faccess_token&oauth_callback%3DMy_URL%26oauth_consumer_key%3DCONSUMER_KEY%26oauth_nonce%3Dfe379af261aca07d890d2cfaa0f19ce0%26oauth_signature_method%3DHMAC-SHA1%26oauth_timestamp%3D1461898452%26oauth_version%3D1.0%26x_auth_mode%3Dclient_auth%26x_auth_password%3DPASSWORD%26x_auth_username%3DEXAMPLE%2540gmail.com

What the error says it expects:

POST&https%3A%2F%2Fwww.instapaper.com%2Fapi%2F1%2Foauth%2Faccess_token&oauth_callback%3DMy_URL%26oauth_consumer_key%3DCONSUMER_KEY%26oauth_nonce%3Dfe379af261aca07d890d2cfaa0f19ce0%26oauth_signature_method%3DHMAC-SHA1%26oauth_timestamp%3D1461898452%26oauth_version%3D1.0%26x_auth_mode%3Dclient_auth%26x_auth_password%3DPASSWORD%26x_auth_username%3DEXAMPLE%2540gmail.com

I pulled my php library from https://github.com/mheap/Instapaper-XAuth-PHP, but it's old so I've tried to modify it to work with the current Instapaper API. I believe I'm generating the signature string correctly and am following the instructions found here: https://dev.twitter.com/oauth/overview/creating-signatures and here: http://oauthbible.com/

I don't know what's wrong with the code, can someone please help?

class XAuth_Connection
{
    private $_headers = array(
        "oauth_signature_method" => "HMAC-SHA1",
        "oauth_version" => "1.0",
        "oauth_callback" => "MY_URL",
        "oauth_consumer_key" => "",
        "oauth_nonce" => "",
        "oauth_timestamp" => ""
    );
    private $_params = array(
        "x_auth_mode" => "client_auth",
        "x_auth_username" => "",
        "x_auth_password" => ""
    );

     private $_access_url = '';

    public function __construct($key, $private, $access_url)
    {
        $this->_headers['oauth_consumer_key'] = $key;
        $this->_headers['oauth_nonce'] = md5(uniqid(rand(), true));
        $this->_headers['oauth_timestamp'] = time();

        $this->_oauth_consumer_private = $private;
        $this->_access_url = $access_url;
    }

    public function set_credentials($user, $password)
    {
        $this->_params['x_auth_username'] = $user;
        $this->_params['x_auth_password'] = $password;
    }

    public function get_params_as_string()
    {

        ksort($this->_params);

        $req = array();
        foreach ($this->_params as $k => $v)
        {
            $req[] = $k ."=". $this->encode($v);
        }

        return implode("&", $req);
    }
    public function get_headers_as_string()
    {

        ksort($this->_headers);

        $req = array();
        foreach ($this->_headers as $k => $v)
        {
            $req[] = $k . "=" . $this->encode($v);
        }

        return implode("&", $req);
    }

    public function generate_signature()
    {

        //combine the parameters, encode, and sort them
        $temp_params = array_merge($this->_params, $this->_headers);

        $encoded_params = Array();
        foreach($temp_params as $k => $v){
            $encoded_params[$this->encode($k)] = $this->encode($v);
        }
        ksort($encoded_params);

        //Build the param string
        $param_base_string = "";
        foreach($encoded_params as $k => $v){
            $param_base_string .= $k .'='. $v . '&';
        }
        $param_base_string = rtrim($param_base_string, '&');

        //create the signature base
        $signature_base = 'POST&' . $this->encode($this->_access_url) .'&'. $this->encode($param_base_string);

        $key = $this->encode($this->_oauth_consumer_private) . '&';

        return base64_encode(hash_hmac("sha1",$signature_base, $key, true));
    }


    public function login()
    {


        $this->_headers['oauth_signature'] = $this->generate_signature();

        ksort($this->_headers);

        $header_str = 'OAuth ';
        foreach ($this->_headers as $k => $v)
        {
            $header_str .= $k.'="'.$this->encode($v).'", ';
        }
        $header_str = rtrim($header_str, ', ');

        $oauth_str = $this->get_params_as_string();

        $ch = curl_init();
        curl_setopt($ch, CURLOPT_URL, $this->_access_url);
        curl_setopt($ch, CURLOPT_POST, true);
        curl_setopt($ch, CURLOPT_POSTFIELDS, $oauth_str);
        curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, false);

        curl_setopt($ch, CURLOPT_HTTPHEADER, array("Authorization: " . $header_str));

        $exec = curl_exec($ch);

        $info = curl_getinfo($ch);
        curl_close($ch);

        if ($info['http_code'] != 200)
        {
            return false;
        }

        parse_str($exec, $r);
        return $r;

    }

    private function encode($s)
    {
        return ($s === false ? $s : str_replace('%7E','~',rawurlencode($s)));
    }

}