From what little I could find it seems they might.
Excerpt from one article.
It was thought that because splake grow more quickly than lake trout and mature and spawn at an earlier age, eventually they would be able to create self-sustaining populations despite lamprey predation.
The first cross between parental species is called an F1, which are the splake hybrids currently stocked in Ontario. However, F2, F3, and F4 crosses, which are second-, third-, and forth-generation hybrids, have also been stocked into the Great Lakes. The province also back-crossed F1 splake with lake trout, searching for a fish with the perfect genetic mix for the Great Lakes. For rehabilitation purposes, the Ministry of Natural Resources (MNR)finally switched back to stocking lake trout during the early 1990s, after lamprey-control programmes proved successful and finding that stocked splake were not propagating or surviving longer than about three years. They were a poor fit for the Great Lakes.
From the MN DNR
September and October
Brook trout and splake are fall spawners, but they cannot successfully spawn in most lakes because they need streams for spawning. Troll or cast for them as they congregate near points.
From the Ont MNR
There is some evidence that where Splake and Brook co-exist splake may compete with Brook Trout for spawning habitat
There has been no indication of succesful natural reproduction in Alqonquin Lakes. It is, however, beleived they atleast go through the spawning act
http://www.mnr.gov.on.ca/MNR/pubs/splake_bio.pdf