As heavier the elements need a high temperature to fuse at a higher the stats may run out of the hydrogen and they collapse leaving the main set of sequence stars and collapses and get hot enough to fuse the helium gas in their cores as carbon.
In such a hypothetical situation the temperature of the sun core would rose up and also allow for the temperature of the score to rise enough with the carbon so as the fusion to take place. Thus the addition of the mass to the sun would be possible.