**Background:** It has long been known that it is relatively consistent with $\mathrm{ZFC + CH}$ that there is no linear ordering $\vartriangleleft $ on a subset $A$ of $\mathbb{R}$  of order-type $\omega_1$ such that $\vartriangleleft$ is projective (when $\vartriangleleft$ is viewed as a subset of $A^2$).  More explicitly, this follows by putting Theorem 2 of Solovay's 1970 paper [A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable][1] together with a classical theorem of Sierpinski (see Andrés Caicedo's answer [here][2] for more detail about Sierpinski's result). 

The above allows me to pose:

**Question.** *What is known about the analogue of the above consistency result, higher up*? More explicitly, I am asking about the status of the above consistency result (relative to appropriate large cardinal axioms) when $\omega$ is replaced by an inaccessible cardinal $\kappa$ and thus $\omega_1$ is replaced by $\kappa^+$, $\mathbb{R}$ is replaced by $\mathcal{P}(\kappa)$, and "projective" is understood as parameterically definable in the natural Kelley-Morse model associated with the model $V_\kappa$ of $\mathrm{ZFC}$, i.e., viewing classes as elements of  $V_{\kappa+1}$.


  [1]: https://chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://people.math.ethz.ch/~fdalio/ZKmodel.pdf
  [2]: https://math.stackexchange.com/questions/33418/does-vitali-construction-exhaust-non-measurable-sets
  [3]: https://chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.ams.org/journals/jams/1989-02-01/S0894-0347-1989-0955605-X/S0894-0347-1989-0955605-X.pdf