According to Archimedes principle the VOLUME of displaced liquid is equal to the submerged object and not WEIGHT. If the liquid is dense enough so that the weight of displaced liquid is greater than that of the object, the object will float if not it will sink.
The article posted talks about volumes of liquid being unequal as grounds for dismissal of Archimedes principle. No such claim about weights of object and displaced liquid being equal was made by Archemides.

Here is the root statement of Archimedes principle:
"Any object, wholly or partially immersed in a fluid, is buoyed up by
a force equal to the weight of the fluid displaced by the object." 
- Source Wikipedia

As you can see the above statement does not say that the weight of displaced liquid is equal to that of the object. Following this statement, Archimedes has worked out the floating or sinking of an object in a given liquid based on the relation between the two weights mentioned in the article. i.e. Archimedes never said those weights were equal and in fact worked out what would happen if the weights were unequal.

In my opinion the above article is arbitrary and baseless. Did all the people in Nepal who had knowledge about physics migrate abroad???