Gespräches or Gesprächs

See title.

Either will do just fine.

Edit: if your boss will accept this authority

I would say Gesprächs.
"Im Laufe eines Gesprächs" sounds better to me.

Did you look it up in Duden?

Are you sure it's good for your wellbeing to be having grammar wars with your boss on a Monday morning, Jayhay?

Thanks to both of you.

As ever, TT is a source for a rapid answer to almost any question.
Long may it thrive.

That's the point, either is correct, it's a question of what you would say. If you can't get your tongue round the "chs" sound then use the form with an "e", whether spoken or written. Forms with an "e" (whether genitive or dative) are older but still acceptable.
I would also say "Gesprächs"

and you would whack the "e" in there (so to speak) if it's sounds really clumsy
for instance, you'd have to say "des Tisches", innit

whatever sounds more natural, really

To me, with the 'e' sounds more natural, as that's what I learned at uni, but in my current job we have to write it without the 'e'.

I like the version with the "E" more as well, but now I know that both are okay ...

Convince him that "Unterhaltung" is the customary term and much more sophisticated than "Gespräch".

many moons ago at school ...under the guidance of a Czech-German dude from Bradford, we were taught with an 's' if the word had 2 or more syllables, 's' alone if it just had one like

...after much German real-world indoctrination I found out both are cool ...same goes for other words like Verkauf or Angriff ..Verkaufes/Verkaufs, Angriffes/Angriffs

more stuff on 's' or 'es'

Spot on with that link, acockreland2balls!

jayhay, you must be shit as an employee, every good employee knows that the boss is always right.