I am not sure what is meant by "run", but I was under the distinct impression that Hong Kong and Shanghai Hotels (aka, the company that owns the Peninsula Hotels chain and a few other things, listed on the HK Exchange) ran at least some of the CX airside lounges. I can confirm this was definitely true at least as recently as 2010.

Update 1: the HKSH website makes it pretty clear they at least run the Wing. From their website, citing achievements for the company in 2011: " A Cathay Pacific business class lounge in the Hong Kong International Airport, managed by Peninsula Clubs & Consultancy Services, was re-opened after extensive renovation." HK Shanghai Hotels website. I suspect if you dig through their annual reports you'll find mentions of what else they manage. Ticker is 45 HK.

Update 2: their 2012 annual report makes it clear they manage both the J and F side of the wing. They mention managing the Cathay Pacific "first and business class lounges". I don't think Sofitel, Plaza Premium or others are managing the CX airside lounges based on this, although it's not definitive that Peninsula manages all. Here's HKSH's annual report, just search for "cathay" in the document. 45 HK 2012 Annual Report

A few years back it was made abundantly clear to me that HKSH managed CX's F and J lounges, including The Cabin, at HKIA. I don't have documentation to prove that but suffice to say the information was from a reliable source.

For those interested, HKSH also manages the Peak Tram and the commercial building it arrives into atop the Peak.