No, the Cogito cannot possibly be proved wrong, even though many philosophers throughout history argued exactly that. However, most philosophers got the Cogito wrong.
There are two main ways philosophers got it wrong in their interpretation of the Cogito, usually to arrive at the irrelevant conclusion that it was not a valid argument.
The first way to get it wrong is indeed to take the Cogito as a logical argument! It is not. The Cogito is not meant for anyone to prove in a logical way the existence of their own mind to somebody else. Indeed, the fact that someone thinks is not apparent to anyone else. Someone saying "I think, therefore I am" merely shows to other people that this person speaks. This in itself could not possibly be a logical proof that the "I" of this person is thinking, let alone that it exists at all. Descartes himself didn't mean the Cogito to be used to assert to the rest of the world the existence of one's own mind.
The other way to get it wrong is to take the Cogito to be about the existence of the self. The self is this psychological phenomenon whereby a subject has a sense of their own biographical identity persisting over time, hopefully throughout life. Although Descartes' discussion of his notion of extreme doubt, leading to the Cogito, is suffused with the notion of self, the notion of self is irrelevant to the Cogito itself.
The Cogito doesn't say "The self thinks, therefore the self exists". This would be a valid argument but one whose premise couldn't be proved true. Instead, the Cogito is expressed in the first-person: I think. And Descartes took the time to explain carefully what was the subject, the "I". The I in the Cogito is the thinking thing. Not the self. Not Descartes. Not a person. Just the thinking thing. The thing doing the thinking.
Thus, the Cogito, if it was a simple logical argument, would be circular: The thinking thing thinks, therefore it exists. But it is not a simple logical argument.
Descartes' Cogito is not a stand-alone metaphysical claim. It is part of Descartes' discussion about body and mind, and more generally Dualism.
He starts by justifying his notion of extreme doubt leading to his conclusion that he is able to doubt even the existence of his own body.
The Cogito, then, is the expression of his conclusion that he cannot similarly doubt the existence of his own mind, because doubting is indeed thinking and thinking implies the existence of the thinking thing, the "I" of the Cogito.
So, as part of this reasoning, the Cogito, rather than a logical argument proving the existence of human minds to the world, is the realisation that one cannot doubt, coherently, the existence of one's own mind.
The Cogito is often misunderstood, and indeed has been often misunderstood throughout history by many philosophers only too anxious to dismiss the Cogito as what they took to be an argument about the existence of the human soul.
You cannot prove the existence of your mind to other people. Whatever you do or say, someone else can reasonably and rationally doubt the existence of your mind. The Cogito doesn't solve this question and wasn't meant to solve this question. Yet, the Cogito is the perfect expression of the fact that no one can coherently doubt the existence of his own mind.
The Cogito is thus a very peculiar kind of argument, and Descartes himself tried to preclude the argument being understood as a logical inference. Something which also explains why the argument has often been misunderstood.
Thinking the Cogito is in effect a performative argument. Any mind thinking the Cogito formally proves its existence to itself. As such it is a fact. Rather than an ordinary argument, it is the expression of a self-demonstrative performance.
I think whenever I perform the act of thinking. Thus, the premise "I think" becomes true, and can be made true, each time I perform the act of thinking. And this, in particular, whenever I think the Cogito itself. Each time I think the Cogito, the premise "I think" becomes true, and with it, the conclusion that I exist.
Bodies don't need to prove to themselves that they exist. If they do, so be it. If they don't, who cares?
The success of the Cogito, possibly the most well-known philosophical argument the world over, comes from its total unassailability in the face of the routine denial of the reality of minds. You can't prove the existence of your mind to somebody else. But whatever other people may say, the Cogito is the expression of the fact that you cannot coherently doubt the existence of your own mind.
This is all that the Cogito is about. The fact that we cannot coherently doubt the existence of our own mind. This is also the force of the Cogito. Nobody would need to pay any attention to Descartes merely asserting the existence of his own mind. Yet, each human being thinking the Cogito will be ipso facto incapable of doubting coherently the existence of their own mind. The mind understood as "the thinking thing".