Seems unlikely, migrating away from an entrenched codec like H264 isn't like a routine software update. It has widespread hardware support, and there's an enormous body of H264 video out there.
As fhn points out, there are now truly open video codecs available (open specification, royalty free, unencumbered by patent terms) that are able to compete with the patent-encumbered ones on technical merit. Seems curious that the patent-holders would want to hike prices in this way and validate the motivation behind the truly open codecs.
Also, the article mentions the licensing fees for H265 were also increased recently. It doesn't seem to give a figure, a quick web search turns up 25% [0] or perhaps 20% [1]. Perhaps I'm missing something obvious but I'm not clear on how the change in price relates to the patent dispute between Nokia and certain laptop manufacturers.
(It seems the H264 fee increase affects streaming providers only, whereas the H265 fee increase did not, as it affected laptop manufacturers.)