The realization question for Rasmussen-invariant vectors

Let $\underline{s},\underline{\vartheta}\in\mathbb{Z}^{\infty}$ be infinite vectors such that, for each vector, all but finitely many entries equal the first entry, and suppose the second entry of $\underline{\vartheta}$ is divisible by four. Rasmussen-invariant realization question. Does there exist a knot $K$ such that

$$(s_0(K),s_2(K),s_3(K),s_5(K),\ldots)=2\underline{s}$$

and

$$(\vartheta_0(K),\vartheta_2(K),\vartheta_3(K),\vartheta_5(K),\ldots)=\underline{\vartheta}?$$

This asks whether the stated divisibility and finite-support restrictions are the only restrictions on these values.