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What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals?

For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > \kappa$ is inaccessible, does $V_\lambda \vDash {}$"$A$ is $\kappa$-homogeneously Suslin"?

The corresponding absoluteness is true for "$\kappa$-universally Baire," so if there are some Woodin cardinals around $\kappa$ then one can prove an approximation to this absoluteness for "$\kappa$-homogeneously Suslin." But such an argument made me wonder if some absoluteness property for "$\kappa$-homogeneously Suslin" might be provable outright.

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  • $\begingroup$ Oops, one has to add the condition that there is a measurable cardinal $\mu$ with $\kappa \le \mu < \lambda$ (we might as well assume that $\kappa$ itself is measurable) or else it is obviously false. (Maybe that's what the down-vote was for?) $\endgroup$
    – Trevor Wilson
    Commented Aug 27, 2013 at 23:54

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