I have the following from a tourism article:
ただ、2018年｛ねん｝に築地｛つきじ｝から豊洲｛とよす｝に市場｛いちば｝機能｛きのう｝が移転｛いてん｝したことをご存知｛ぞんじ｝の方｛かた｝もいらっしゃるでしょう。「築地｛つきじ｝は今｛いま｝も観光｛かんこう｝に値｛あたい｝するスポットなの？」。そんな疑問｛ぎもん｝を持｛も｝つ方｛かた｝もいるかもしれません。
In the first case, it uses いらっしゃる, a respectful version of the second case's いる.
Yet, both uses 方{かた} as a respectful way to refer to a person, so I assume that the general nuance of the sentences is respectful. If that's the case, why doesn't the second case use いらっしゃる also?
USUALLY you'd have to stick to either 丁寧語 (polite word) or 普通語 (casual word) in the whole context. I say USUALLY because sometimes even Japanese people ignore this rule. So yeah you are absolutely right, it should have been "そんな疑問を持つ方もいらっしゃるかもしれません。" if whoever wrote these sentences had followed the rule. In saying that, I don't think そんな疑問ぎもんを持もつ方かたもいるかもしれません。 is totally incorrect either.
There is no important reason. This is not a formal business letter, and the writer does not have to be very polite to their readers. The use of いらっしゃる is purely optional in this context. Maybe the writer thought repeating いらっしゃる would look monotonous or an overkill, but probably the writer was totally unaware of the "inconsistency".
Related: What is the difference between 「しております」and 「しています」?
