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Do Bnei Keturah need a Bris upon conversion according to Rambam?

Rambam paskens (Mishneh Torah, Kings and Wars 10:7-8):

>הַמִּילָה נִצְטַוָּה בָּהּ אַבְרָהָם וְזַרְעוֹ בִּלְבַד. שֶׁנֶּאֱמַר (בראשית יז, ט) "אַתָּה וְזַרְעֲךָ אַחֲרֶיךָ". יָצָא זַרְעוֹ שֶׁל יִשְׁמָעֵאל שֶׁנֶּאֱמַר (בראשית כא, יב) "כִּי בְיִצְחָק יִקָּרֵא לְךָ זָרַע". [...] אָמְרוּ חֲכָמִים שֶׁבְּנֵי קְטוּרָה שֶׁהֵם זַרְעוֹ שֶׁל אַבְרָהָם שֶׁבָּא אַחַר יִשְׁמָעֵאל וְיִצְחָק חַיָּבִין בְּמִילָה. וְהוֹאִיל וְנִתְעָרְבוּ הַיּוֹם בְּנֵי יִשְׁמָעֵאל בִּבְנֵי קְטוּרָה יִתְחַיְּבוּ הַכּל בְּמִילָה בַּשְּׁמִינִי.‏
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>Circumcision was commanded only to Avraham and his offspring, as it is written, "You and your offspring after you." The offspring of Yishmael are excluded, as it is written, "For through Yitzchak will your offspring be called." [...] Our Sages said that the children of Keturah, the offspring of Avraham who came after Yishmael and Yitzchak, are obligated in circumcision. Since nowadays the children of Yishmael and Keturah are mixed together, they are all obligated in circumcision on the eighth day.

According to this position, suppose a descendant of Keturah was circumcised properly, as he is supposed to, and later wished to convert to Judaism. Would he need a new circumcision (i.e. "drawing the blood of circumcision") as a part of his conversion, or can he rely on his previous circumcision, which was done in fulfillment of the mitzvah? 

Muslims observe circumcision, albeit at thirteen years instead of eight days. Suppose a Muslim was previously circumcised, which according to Judaism he'd be obligated to do as a potential descendant of Keturah. If the answer to the first question is no, would a Muslim similarly be exempt, or since he is moving from a doubtful obligation to a definite one, he would be obligated in a second circumcision out of a doubt (but would undergo the circumcision without a blessing)?