Does an ordered abacus have a higher mass (no matter how small) than a random one? 
In 2019, physicist Melvin Vopson of the University of Portsmouth proposed that information is equivalent to mass and energy, existing as a separate state of matter, a conjecture known as the mass-energy-information equivalence principle. This would mean that every bit of information has a finite and quantifiable mass.

So, if I invest energy in creating an ordered pattern on an abacus, it will have a different mass than a random one. Now the marbles making up the pattern stays the same. But obviously their positions wrt one another are different. The gravitational potential, however small, is different.
Is this plausible? Does an ordered abacus weigh more? Could we, in principle, weigh two identical abacusses and determine by weight which is more ordered?
 A: The answer to your question is "almost certainly not", for 3 reasons:
(1) First, and probably most importantly, the paper you reference is highly speculative. We don't really understand the connection, if any, between information and energy, and if there is a relation it may well be more complicated than "more information = more energy".
(2) Secondly, the information contained in the arrangement of a few dozen beads is utterly dwarfed by the information contained in the arrangement of the $10^{25}$ or so atoms making up the beads.
(3) Lastly, even if the first two items were somehow overcome, the ordered arrangement of beads has less information than a random arrangement. This may seem counter-intuitive at first, but it becomes more obvious if you think about how much information is required to describe the beads. For an ordered arrangement you might be able to specify a simple rule (like "all the beads pushed as far to the left as possible"). For a truly random arrangement you would need, in general, to describe the position of each bead separately.
Similarly, in computer science a random string cannot be compressed, whereas ordered strings (like text) can generally be compressed, and the more ordered they are the more they can be compressed.
A: 
So, if I invest energy in creating an ordered pattern on an abacus, it will have a different mass than a random one.

No The energy invested will be used as work to change the potential energy of each bead by changing its height wrt to the bottom of the abacus ($m_{bead}gh$). None of this energy is being transformed into mass. You can argue (since gravity decreases with height) that the weight of an abacus with all its beads against the top will weigh ($W = m_{beads}g(h)$ ) slightly less than an abacus with its beads against the bottom and use this to state that in general the weight of an abacus will be a function of positions of the beads but it won't be a function of "order"
