Yes, we support the export of pricelist items to specific product prices in Prestashop. This functionality has some limitations because of big difference between Odoo Pricelist functionality and Prestashop specific prices functionality.
Initial import of Prestashop specific prices to Odoo Priceslists
Video that is showing this initial import process. Note that this step is assumed to be done
Export periodically changes to pricelist from Odoo to Prestashop
Note that pricelists export is done not in real time, but is periodically synced by special “Scheduled Action“ with name “Integration: Send Pricelist Items“. As shown on screenshot below it is running every day.
Note that this schedule action checks below checkbox on the product. And only if it is set, it will send updates for special prices for this product to Prestashop. This checkbox is automatically checked when pricelist item related to this product is updated. So scheduled action just checking all products with this checkbox.