Projective well-ordered sets, higher up

Question

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 together with a classical theorem of Sierpinski (see Andrés Caicedo's answer here 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 $V_{\kappa+1}$.

Answer 1

The theory $T_0$ = ZFC + "there is an inaccessible $\kappa$ such that every wellorder of a subset of $V_{\kappa+1}$ which is definable over $V_{\kappa+1}$ from parameters has length $<\kappa^+$" is equiconsistent with $T_1=$ ZFC + "there are two inaccessible cardinals". These are in fact also equiconsistent with $T_0^+$, where this is like $T_0$, but we allow any wellorders definable over $V$ from parameters in $V_{\kappa+1}$.

Proof: Assuming $T_0$ holds, note that $\kappa$ and $\kappa^+$ are both inaccessible in $L$. (Otherwise $\kappa^+$ is a successor cardinal in $L$, say $\kappa^+=\gamma^{+L}$, but then from a set $A\subseteq\kappa$ which codes a wellorder of length $\gamma$, we can define over $V_{\kappa+1}$ a wellorder of some subset of $\mathcal{P}(\kappa)$ of length $\gamma^{+L}$, using the usual $L$-definability.

Now suppose $T_1$ holds, and let $G$ be $V$-generic for the Levy collapse making $\lambda=\kappa^{+V[G]}$. Then $V[G]$ satisfies $T_0$, as witnessed by $\kappa$. Then $V_\kappa^{V[G]}=V_\kappa$ and $\kappa$ is still inaccessible in $V[G]$ (since the forcing is ${<\kappa}$-closed). Now let $A\subseteq\kappa$ with $A\in V[G]$, and let $W$ be a wellorder of some subset of $\mathcal{P}(\kappa)\cap V[G]$ which is definable over $V[G]$ from the parameter $A$. There is $\beta<\lambda$ such that $A\in V[G\upharpoonright\beta]$. Now I claim that every $X\subseteq\kappa$ which is in the field of $W$, is in $V[G\upharpoonright\beta]$, and so the length of $W$ is $<\lambda=\kappa^{+V[G]}$. This is just the usual argument using the homogeneity of the factor forcing producing $G\upharpoonright[\beta,\lambda)$. (So not only are the wellorders definable over $V_{\kappa+1}^{V[G]}$ from parameters of short length, in fact all those definable over $V[G]$ from parameters in $V_{\kappa+1}^{V[G]}$ are of short length.)

This model was used by Philipp Schlicht for a related construction in "Perfect subsets of generalized Baire spaces and long games" (a preprint can be found here).