Reinit filter product type with the latest value

[monsieurtanuki]

Problem

The new product type filter has 4 options (OxF).
It's displayed in the search page.
We don't store the latest selected option of this filter.
Next time we go to the search page, the selected option is always "food", though we can expect the value to be the latest.

Proposed solution

Store the latest selected option in the preferences, and reuse it next time we land on the search page.

Part of

• Add a project switcher to search (and store the type on each product opened "off", "obf", "opf" and "opff", and for any search we store locally) #5586
• feat: 5586 - OxF filter for term searches #5637