patch.js

// replace your start date instance onfirstselect event with this:

onfirstselect: function (instance, start) {
    startDate = start.clone();
    startInstance.globals.startSelected = false;
    startInstance.hideDropdown();
    endInstance.showDropdown();
    endInstance.setDisplayDate(start.clone());
    endInstance.config.minDate = startDate.clone();
    endInstance.config.startDate = startDate.clone();
    endInstance.config.endDate = null;
    endInstance.globals.startSelected = true;
    endInstance.globals.endSelected = false;
    endInstance.globals.firstValueSelected = true;
    if (endDate && startDate.isAfter(endDate)) {
        [startDate, endDate] = [endDate, startDate];
        endInstance.config.endDate = endDate.clone();
    }
    fillInputs();
    endInstance.updateHeader();
    endInstance.reDrawCells();
    endInstance.updateTimePickerDisplay();
}