TITLE: Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?
QUESTION [15 upvotes]: Let $K$ be a knot in the 3-sphere $S^3$.
Here we  denote by $s(K)$  Rasmussen's s-invariant for $K$, 
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a $2$-handle along $K$ with framing $n$.
My question is the following.

For two knots $K$ and $K'$ such that  $X_{K}(0)$ and  $X_{K'}(0)$ are diffeomorphic,
  is it true that $s(K)=s(K')$?

I am also interested in the same question for Ozsváth-Szabó's $\tau$-invariant. 
Note that it is known that there exist knots $J$ and $J'$ such that 
 $\partial X_{J}(0)$ and  $\partial X_{J'}(0)$ are diffeomorphic,
and   $s(J) \neq s(J')$, see Yasui's paper Corks, exotic 4-manifolds and knot concordance.

REPLY [8 votes]: Lisa Piccirillo just posted a few days ago this paper on the arXiv. Corollary 1.3 asserts exactly that $s$ is not a 0-trace invariant of the knot.