NO. No one on Earth can claim to "own the truth" -- not even the natural sciences. And mathematics has no anchor on Nature.
With physics, the elusive truth becomes the object itself, which physics trusts using the scientific method, as fairly as humanly possible and as objectively (friend and foe) as possible.
With mathematics, on the other hand, one must trust using only logic, and the most amazing thing has been how much the Nature as seen by physics (the Wirklichkeit) follows the logic as seen by mathematics (without necessarily using Wirklichkeit) -- and vice-versa. This implies that something is true in Wirklichkeit iff (if and only if) it is logical.
Also, any true rebuffing of a "fake controversy" (i.e., fake because it was created by the reader willingly or not, and not in the data itself) risks coming across as sharply negative. Thus, rebuffing of truth-deniers leads to ...affirming truth-deniers. The semantic principle is: before facing the night, one should not counter the darkness but create light. When faced with a "stone thrown by an enemy" one should see it as a construction stone offered by a colleague.
But everyone helps. The noise defines the signal. The signal is what the noise is not. To further put the question in perspective, in terms of fault-tolerant design and CS, consensus (aka,"Byzantine agreement") is a design protocol to bring processors to agreement on a bit despite a fraction of bad processors behaving to disrupt the outcome. The disruption is modeled as noise and can come from any source --- attackers or faults, even hardware faults.
Arguing, in turn, would risk creating a fat target for bad-faith or for just misleading references, exaggerations, and pseudo-works. As we see rampant on RG, even on porous publications cited as if they were valid.
Finally, arguing may bring in the ego, which is not rational and may tend to strengthen the position of a truth-denier. Following Pascal, people tend to be convinced better by their own-found arguments, from the angle that they see (and there are many angles to every question). Pascal thought that the best way to defeat the erroneous views of others was not by facing it but by slipping in through the backdoor of their beliefs. And trust is higher as self-trust -- everyone tends to trust themselves better and faster, than to trust someone else.
What is your qualified opinion? This question considered various options and offers a NO as the best answer. Here, to be clear, "truth-denial" is to be understood as one's own "truth" -- which can be another's "falsity", or not. An impasse is created, how to best solve it?