Imho

That is, an objection has been made during the deposition to the questions. When an objection is made during a deposition, it noted and the person being deposed may be asked to answer the question nevertheless.

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Note added at 27 mins (2013-01-11 20:08:40 GMT)
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oops.... "it may be noted and..."

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Note added at 1 hr (2013-01-11 20:56:14 GMT)
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The following is related, I think:

En ce qui concerne les objections prises sous réserve, le législateur doit prévoir la possibilité de les prendre en note dans un document distinct qui serait gardé sous scellés jusqu’à l’audition. Ainsi, les parties ne pourraient utiliser l’information qu’une fois l’objection tranchée. Il en est de même pour le juge, qui ne pourrait prendre connaissance de la réponse à la question sous objection qu’après sa décision. À cet égard, le Barreau souligne l’importance du rôle du sténographe qui pourra être le gardien du document sous scellés et qui assurera l’intégrité du processus.
https://www.barreau.qc.ca/pdf/medias/positions/2012/20120202...

The objected is noted in the deposition when it is taken. The judge, absent during the deposition, may only rule on the objection at a later date.

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Note added at 3 hrs (2013-01-11 23:06:08 GMT)
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A note (for Nikki): "Under objection" may be infrequent (as you note in the Answer you proposed, but that is also true for "sous objection." ;-)

Another note (for writeaway) re: "not legal(see)":

Form "Appeal System Practice & Procedures":
5. ALTERNATIVE DISPUTE RESOLUTION [page 2]
• To recognize the diversity of issues appealed, a variety of dispute resolution approaches are made available by the Appeals Branch (AB). The approach chosen is geared to the nature of the issues under objection.
[...]
2. Review Stage
(c) Appeals Resolution Officer Contact [p. 10]
• The ARO reviews the file to determine the facts, the nature of the issues under objection and the identity of the participating parties.
[more instances later in the doc]
http://www.wsib.on.ca/files/Content/AppealsAppealSystemPract...

Government of Ontario: Ministry of Finance
Notice of Objection
Issues, Facts and Reasons - Please describe each issue under objection and include the specific dollar amount for each issue in dispute.
http://www.forms.ssb.gov.on.ca/mbs/ssb/forms/ssbforms.nsf/Ge...$File/TXT_1406E.htm

From the "Atlantic Reporter: : Cases Argued and Determined in the Courts of Connecticut, Delaware, Maryland, New Hampshire, New Jersey, Pennsylvania, Rhode Island, Vermont," Volume 82 (Google eBook):
The answer was read under objection and exception;
http://preview.tinyurl.com/a98g8jd

Likely to work in AmE/CanE/BrE, but not in Turkish.