Surjectivity conjecture for the field-difference homomorphism

Let $\mathcal{C}$ be the smooth concordance group, and for each prime $p$ define

$$\Sigma_p(K)=\frac{s^{\mathbb{Q}}(K)-s^{\mathbb{F}_p}(K)}{2}.$$

Writing $P$ for the set of primes, let

$$\Sigma\colon\mathcal{C}\longrightarrow\bigoplus_{p\in P}\mathbb{Z}$$

be the homomorphism whose $p$-component is $\Sigma_p$.

Surjectivity conjecture. The homomorphism $\Sigma$ is surjective.

This asks whether the differences between the Rasmussen invariants over characteristic zero and over prime fields can realize every finitely supported integer sequence. The source provides no resolution of this question.