Hi,

You can set store=True for compute functions then the value will be stored in the database.
Here you can redefine the sale order count field with store=True and also give sale_order_ids as depends on the compute function.

ie,

sale_order_count = fields.Integer(compute="_compute_sale_order_count",
                                      string="Sale Order Count",
                                      store=True)
    @api.depends('sale_order_ids')
    def _compute_sale_order_count(self):
        # retrieve all children partners and prefetch 'parent_id' on them
        all_partners = self.with_context(active_test=False).search(
            [('id', 'child_of', self.ids)])
        all_partners.read(['parent_id'])

        sale_order_groups = self.env['sale.order'].read_group(
            domain=[('partner_id', 'in', all_partners.ids)],
            fields=['partner_id'], groupby=['partner_id']
        )
        partners = self.browse()
        for group in sale_order_groups:
            partner = self.browse(group['partner_id'][0])
            while partner:
                if partner in self:
                    partner.sale_order_count += group['partner_id_count']
                    partners |= partner
                partner = partner.parent_id
        (self - partners).sale_order_count = 0

Regards