The CS1W-CN226 is a RS232 serial cable.
It is designed to work with a serial port on the computer.
If the conversion cable you are referring to is a USB to serial
adapter, it may or may not work. Please search this forum
for many discussions regarding these devices.
Omron offers the CS1W-CIF31 USB to serial adapter that they have
tested and will support.
I hope this will help.