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Problem 598 concerns all countably infinite subsets and witnesses of a prescribed cardinal size. The Magidor relation in Garti–Hayut Claim 1.10 instead concerns bounded countably infinite subsets of a singular cardinal. The following author-recorded argument supplies the domain bridge. It does not independently verify the forcing theorem or the other half of the proposed independence result.

Let $\operatorname{cf}(\lambda)=\omega$ and let $\kappa<\lambda$ be uncountable and regular. Suppose $\lambda\to[\lambda]^{\aleph_0\text{-bd}}_\kappa$; the domain includes all countable order types. For a coloring $f:[\lambda]^{\aleph_0}\to\kappa$, restrict to bounded subsets and choose $A\subseteq\lambda$ of size $\lambda$ and a color $i$ omitted there. Choose an increasing cofinal sequence $(\lambda_n)_{n<\omega}$ in $\lambda$. Some $A\cap\lambda_n$ has size at least $\kappa$, since otherwise its countable union would have size below the uncountable regular cardinal $\kappa$. Take $X\subseteq A\cap\lambda_n$ of size $\kappa$. Every countable subset of $X$ is bounded in $\lambda$, so it omits $i$. Consequently

$$\lambda\to[\kappa]^{\aleph_0}_\kappa.$$

Color monotonicity also needs no additional set theory. If the bounded omitting relation holds with $\rho$ colors and $\rho\le\kappa$, compose a $\kappa$-coloring with a surjection onto $\rho$. Omitting a $\rho$-color omits its nonempty fiber of $\kappa$-colors.

In Claim 1.10(a) of arXiv:1801.00239v3, take $\beta=1$. The theorem starts from I1 and produces a Magidor $\lambda$ with $\alpha_M(\lambda)=\aleph_2$. Its proof first uses Prikry forcing at a measurable $\mu<\lambda$, then a countably closed Lévy collapse. Using the standard facts that these steps add no reals, and the theorem's preservation of $\lambda$ as a limit cardinal, the final continuum is below $\lambda$: the unchanged reals inject into the old ordinal $\mu<\lambda$. Hence the final $\kappa=\mathfrak c^+$ is below $\lambda$ and at least $\aleph_2$. Monotonicity and the domain bridge give the desired negative instance at $m=\lambda$. No assumption of CH is needed for this specialization.

The source introduction and Claim 1.10's proof paragraph supply the domain convention and the two forcing steps. Preservation of Magidority, the no-new-reals facts, and the supporting forcing lemmas remain imported premises, not reconstructed proofs here. The independent scrutiny needed for the whole conclusion is separate from checking this bridge.

For the positive-in-$L$ half, Wu's existing claim uses a finite-subset coding and elementary-substructure equivalence followed by a condensation-to-$0^\sharp$ theorem. Those textbook interfaces are unverified in this note. Checking their exact hypotheses, and the final cardinals in the negative model, is the next useful source test. A missing model-theoretic hypothesis or forcing-preservation clause leaves the whole independence conclusion conditional on that gap.