This is a common writing mistake, but the object and pronoun need to match in singularity or plurality.
Incorrect: Baby Silas is really enjoying their bath.
Correct: Baby Silas is really enjoying his bath.
A related mistake is when a single subject is paired with a plural verb. Again, the subject and verb need to match in singularity and plurality.
Incorrect: The president and the rest of his staff is spending the afternoon at Camp David.
Correct: The president and the rest of his staff are spending the afternoon at Camp David.
Something to note is that gender-neutral pronouns (e.g. zie, zim, zir, zis) are growing in use. However, they’re not currently the norm in the majority of business writing and correspondence. If a client, colleague or audience prefers the use of gender-neutral pronouns, be sure to learn more information about how to use them. Also, when writing, clarify that you’re using gender-neutral pronouns to your audience. Don’t assume they already know.