Proof.
In the base case that \(k = 0\), this factorization system is recorded as [Lur09, Ex. 5.2.8.16], which is easy to check is compatible with the Cartesian symmetric monoidal structure and generated by the single morphism \(\{ S^{n+1} \rightarrow{\sf pt}\}\). So, let us assume that \(k > 0\).
If \(n = -2\), then this is the trivial factorization system \((\mathrm{Cat}_{(\infty, {k})},\mathrm{Cat}_{(\infty, {k})}^\simeq)\) (as in Example B.1.7). Moreover, it is trivially compatible with the Cartesian symmetric monoidal structure, and it is also clearly generated by the set \(\{ \partial c_i \rightarrow c_i \}_{0 \leq i \leq k}\): by induction (and the universal property of the categorical suspension functor \(\Sigma[-]\)), a morphism \(\mathcal C\rightarrow\mathcal D\) is right orthogonal to this set if and only if it is an equivalence on maximal subgroupoids and on hom-\((\infty,k-1)\)-categories.
From here, in the case that \(n > -2\) (and \(k > 0\)) the claim follows by applying Theorem B.4.1 inductively (varying both \(k\) and \(n\) simultaneously). ◻