Presently I've done no goodly search to write what I'll write here. Here I am going to write about an issue of English grammar, but its purpose is neither for informative article nor for sharing knolwedge. It's only for this diary-like writing exercise of mine. This is only a memo, just a memo, neither more nor less than a memo.

That having said, ...

In English, what pronoun should I use when I don't know if the one in question is male or female?

Some seem to use always either "he" or "she." Some may use "they" in the place of a third-person singular pronoun. Some seem to use "s/he." But it is nothing but my impression; for English, I'm still just a learner. Probably I should consult a grammar book or something, or ask the question on a website designed for such a purpose.

Be that as it may, now a new suspicion has occurred to me. Is it not particularly necessary to use a fixed third-person singular pronoun for this use? That is, as to which pronoun you take, can you choose either arbitrary depending on what image you could have then and there? And even in a mixed way?

That is, can I write as follows?

'A system engineer would discuss with "his" customer how to design the system and a hair stylist would tell "her" customer how to maintain "their" hair style.'

In this example, when I wrote the sentence, I personally imagined a male system engineer and a female hair stylist, so I used respective pronouns according to this imagination.

Can such a sentence sound natural? Or at least is such a sentence be grammatically acceptable...?