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Citing (almost) A question about ordinal definable real numbers

If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is added to it: "There exists a denumerably infinite and ordinal definable set of real numbers, not all of whose elements are ordinal definable"

1) If we drop of real numbers then, as mentioned elsewhere, a positive answer readily comes in the form of $S=\{X,Y\}$ in the Sacks$\times$Sacks extension $L[x,y]$ of $L$, where $X$ is the $L$-degree of the real $x$ and $Y$ is the $L$-degree of the real $y$, so $X,Y$ are sets of reals.

2) As Ali Enayat explained elsewhere, a symmetric extension of $L$ on the base of a Jensen 1970 forcing yields a positive answer with the key absence of AC: it gives a model of ZF.

In the reference above, some approaches (eg, based with the Mathias forcing) were outlined. Has there been any further development possibly with a full answer of the original question?

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    $\begingroup$ Oops, I tried to just vote to close as duplicate, but somehow things have gone wrong. The message that the question is answered on the other question is of course not really accurate, since that answer doesn't yet actually answer the question. $\endgroup$
    – Joel David Hamkins
    Commented Jul 30, 2014 at 16:59
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    $\begingroup$ I see; it seems that I have unanticipated powers! The question is definitely not answered, although there is of course useful information in Andres's post over there. I think this message probably should be changed network-wide, since I expect that there would be a similar issue arising whenever the original question has answers posted but none accepted. $\endgroup$
    – Joel David Hamkins
    Commented Jul 30, 2014 at 17:21
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    $\begingroup$ The auto-message above even instructs us to "ask a new question", which to my understanding is not the recommended procedure. $\endgroup$
    – Joel David Hamkins
    Commented Jul 30, 2014 at 17:23
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    $\begingroup$ The wording of the message is such because one is indeed not supposed to mark something as a duplicate of a question without answers, and the system will normally not even allow it (IIRC). So the problem is not the message, the problem is the policy (whose rationale I never really understood). $\endgroup$
    – Emil Jeřábek
    Commented Jul 30, 2014 at 17:55
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    $\begingroup$ But even if that is the policy, there is still a problem with the message, which is that the message says that the original question has been answered, but this is not true. The question has merely had an answer posted, but the question remains unanswered. The further message that "if those answers do not fully address your question, please ask a new question" is even more problematic and should definitely be changed. $\endgroup$
    – Joel David Hamkins
    Commented Jul 30, 2014 at 18:54

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