RJ45 to RJ11 Fax Lines

In our new building the contractor who ran the data cabling installed the fax lines using regular CAT6 cable with an RJ45 jack. One of our departments had a fax machine with an RJ45 fax port and it works just fine, but is it possible for our other machines that have RJ11 ports to somehow use the RJ45 wall jack?

I've found some things like this: http://www.amazon.com/Gino-Telephone-RJ11-Connector-Cable/dp/B006OIDURW and guides on how to make the cable yourself, but there are always people in comments insisting that it wouldn't work. So what do you guys think, could I make it work by crimping an RJ11 end onto a CAT6 cable, solder something together, or will it just not work?