According to the actual paper (https://arxiv.org/pdf/2506.24088), it has been an open conjecture since at least 1977. The quote:

> Unknotting number has long been conjectured to be additive under connected sum; this conjecture is implicit in the work of Wendt, in one of the first systematic studies of unknotting number [37]. It is unclear when and where this was first explicitly stated; most references to it call it an ‘old conjecture’. It can be found in the problem list of Gordon [13] from 1977 and in Kirby’s list [16].

'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) = u(K1) + u(K2). It is this that has (assuming the paper is correct) been proven false. A deceptively simple property!

edit: I initially incorrectly had a ≤ sign instead of =

> 'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) ≤ u(K1) + u(K2).

Kinda like the triangle inequality[1] of knots?

I recall the triangle inequality was useful for several cases in Uni, if so I guess I can see it might be a similarity useful inequality in knot theory.

[1]: https://en.wikipedia.org/wiki/Triangle_inequality