Custom Post Type pagination when CPT ‘rewrite’ rule and a page have the same slug

I would like to achive the following link structure on my site:

1. /properties/ -> a properties page with properties CPT, page 1.
2. /properties/page/2 -> a properties page with properties CPT, page 2.
3. /properties/property-name -> a single properties page with a property details.

I have a Custom Post Type (CPT) called property. It’s registered as following with the rewrite rule specified as 'rewrite' => array('slug' => 'properties', 'with_front' => false):

function property_post_type_init() {
  $labels = array(
    'name' => _x('Properties', 'post type general name'),
    'singular_name' => _x('Property', 'post type singular name'),
    'add_new' => _x('Add New', 'property')
  );
  $args = array(
    'labels' => $labels,
    'public' => true,
    'publicly_queryable' => true,
    'show_ui' => true,
    'rewrite' => array('slug' => 'properties', 'with_front' => false),
    'query_var' => true,
    'capability_type' => 'post',
    'hierarchical' => false,
    'show_in_nav_menus' => false,
    'menu_position' => 5,
    'supports' => array(
      'title',
      'editor',
      'revisions'
    )
  );
  register_post_type('property',$args);

I also have a page with the slug properties and the permalink structure is set as /%category%/%postname%/

On the Properties page I construct a new WP_Query object with the following arguments:

$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
$args = array(
    'post_type' => 'property',
    'orderby' => 'date',
    'order' => 'DESC',
    'paged' => $paged
);
return $args;

I use a custom pagination code (in functions.php) which generates pagination correctly with the correct urls:

<my-site>/properties/
<my-site>/properties/page/2/
<my-site>/properties/page/3/

However, when I click on any page in the pagination I get redirected to 404 page. I’ve expected the query params and they are the following:

[page] => 2
[property] => page
[post_type] => property
[name] => page

Which is not correct! I don’t understand where these parameters are coming from:

[page] => 2
[property] => page
[name] => page

I’m sure it’s something to do with the rewrite rule when registering a CPT.

How can I achive the following set up to work correctly:

1. A properties page that displays all properties custom post types posts with pagination.
2. A properties custom post type with 'rewrite' => array('slug' => 'properties', 'with_front' => false) rule.
3. Permalink of /%category%/%postname%/ structure

So that I have the following link structure:

1. /properties/ -> a properties page with properties CPT, page 1.
2. /properties/page/2 -> a properties page with properties CPT, page 2.
3. /properties/property-name -> a single properties page with a property details.

Hope that makes sense.

Thanks,
Dasha