When Ester was in the harem, and presumably after she became queen, Mordechai was constantly around to see how she was doing.
וּבְכָל־י֣וֹם וָי֔וֹם מָרְדֳּכַי֙ מִתְהַלֵּ֔ךְ לִפְנֵ֖י חֲצַ֣ר בֵּית־הַנָּשִׁ֑ים לָדַ֙עַת֙ אֶת־שְׁל֣וֹם אֶסְתֵּ֔ר וּמַה־יֵּעָשֶׂ֖ה בָּֽהּ׃ (2:11)
בַּיָּמִ֣ים הָהֵ֔ם וּמָרְדֳּכַ֖י יֹשֵׁ֣ב בְּשַֽׁעַר־הַמֶּ֑לֶךְ וגו' (2:21)
The thing is, Mordechai himself told her not to reveal her nationality.
לֹא־הִגִּ֣ידָה אֶסְתֵּ֔ר אֶת־עַמָּ֖הּ וְאֶת־מֽוֹלַדְתָּ֑הּ כִּ֧י מָרְדֳּכַ֛י צִוָּ֥ה עָלֶ֖יהָ אֲשֶׁ֥ר לֹא־תַגִּֽיד׃ (2:10)
However, if Mordechai, who was presumably known to be the leader of the Jewish community, was constantly wondering about her, wouldn't that be suspicious? People might figure out that they're related.
What did Mordechai claim to be doing when he was checking in on her, and staying by the king's gate?