Restricting searches to categories in Woocommerce

Firts of all, thanks for the great plugin, but I have a small issue. I am using Woocommerce and I would like to search for products in different categories. I made this search form:

<form method="get" action="<?php echo esc_url( home_url( '/'  ) ); ?>">
    <div><label class="screen-reader-text" for="s">Search</label>
    <input type="text" value="<?php the_search_query(); ?>" name="s" id="s" />
	<?php
		wp_dropdown_categories(array(
			'name' => 'product_cat',
			'show_option_all' => 'Select a category',
			'taxonomy' => 'product_cat'));
	?>
    <input type="submit" id="searchsubmit" value="Search" />
	<input type="hidden" name="post_type" value="product" />
    </div>
</form>

1. When there is no search word and no category assigned, all products are returned.
2. When I assign a category in the dropdown and no search word, 404 page is returned.
3. Search by keyword works fine, but no matter what category I assign.

In Relevanssi I chose to index product and product_cat and Default operator for the search is AND. I am sure I am missing something I can’t figure what.

https://www.ads-software.com/plugins/relevanssi/