The news story appears to have originally come from New Scientist, and they apparently quote a figure of 7 mm/year (rather than 7 to 10), which they attribute to 3mm/year from annual sea level rise and the extra 4mm/year to the effect of supposedly enhanced trade winds.
The disappearing islands are in the Solomon Islands group, which I understand to be an area of the world which is frequently affected by earthquakes. My personal view is that you can't claim sea level rise as being due to climate change in an earthquake-prone part of the world. You'd have to at least look at an island in a part of the world which only sees low seismic activity.
To illustrate what I mean, consider the major earthquake that caused a tsunami and affected Japan in March 2011 (which is known officially as the Great East Japan Earthquake or Tohoku Earthquake). This news article gives some idea how much ground level drop occurred on the East coast of Japan:
The maximum ground level drop was 84 cm recorded in Rikuzentakata city. The ground level drop, equivalent to an effective sea level rise, has not been widely reported in Western countries, but the lack of reporting of the issue is justifiable as this particular effect of the earthquake is almost trivial compared with the tsunami event itself.
The sea level rise range predicted by the latest IPCC report AR5 are 28 to 98 cm by the year 2100 (after checking the figures on the RealClimate blog). So in an earthquake-prone part of the world, you could see the entire end-of-century IPCC rise literally the next day if a major earthquake happens to occur. Also the ground may be slowly shifting vertically between earthquake events in these earthquake-prone areas.