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Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals and reals is Baire measurable.

My question is:

In HOD, is every set definable from ordinals and reals (in HOD) Baire measurable? (i.e. is the property that a definable set is Baire measurable preserved by passage from $N$ to HOD?). If we consider also HOD(HOD) does the same remain true?

Edit: I rephrase my question to be: is there a model of set theory $M$ such that in HOD$^{M[G]}$ every set of reals definable from ordinals and reals is Baire measurable?

Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals and reals is Baire measurable.

My question is:

In HOD, is every set definable from ordinals and reals (in HOD) Baire measurable? (i.e. is the property that a definable set is Baire measurable preserved by passage from $N$ to HOD?). If we consider also HOD(HOD) does the same remain true?

Edit: I rephrase my question to be: is there a model of set theory $M$ such that in HOD$^{M[G]}$ every set of reals definable from ordinals and reals is Baire measurable? Can we find a model of set theory in which HOD, HOD(HOD) etc.. all satisfy the same property above?

Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals and reals is Baire measurable.

My question is:

In HOD, is every set definable from ordinals and reals (in HOD) Baire measurable? (i.e. is the property that a definable set is Baire measurable preserved by passage from $N$ to HOD?). If we consider also HOD(HOD) does the same remain true?

Edit: I rephrase my question to be: is there a model of set theory $M$ such that in HOD$^M[G]$ every set of reals definable from ordinals and reals is Baire measurable?

Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals and reals is Baire measurable.

My question is:

In HOD, is every set definable from ordinals and reals (in HOD) Baire measurable? (i.e. is the property that a definable set is Baire measurable preserved by passage from $N$ to HOD?). If we consider also HOD(HOD) does the same remain true?

Edit: I rephrase my question to be: is there a model of set theory $M$ such that in HOD$^{M[G]}$ every set of reals definable from ordinals and reals is Baire measurable?

Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals and reals is Baire measurable.

My question is:

In HOD, is every set definable from ordinals and reals (in HOD) Baire measurable? (i.e. is the property that a definable set is Baire measurable preserved by passage from $N$ to HOD?). If we consider also HOD(HOD) does the same remain true?

Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals and reals is Baire measurable.

My question is:

In HOD, is every set definable from ordinals and reals (in HOD) Baire measurable? (i.e. is the property that a definable set is Baire measurable preserved by passage from $N$ to HOD?). If we consider also HOD(HOD) does the same remain true?

Edit: I rephrase my question to be: is there a model of set theory $M$ such that in HOD$^M[G]$ every set of reals definable from ordinals and reals is Baire measurable?