Before I answer I wish to state that I am a proud supporter of the LGBTQ+ community and don't particularly care what pronoun anyone prefers or chooses for themself, as long as it makes them feel comfortable and respected.
With the risk of seeming a bit flippant in my answer: No, there is NOT a correct gender-neutral singular pronoun.
And, in support of my answer, I point to all of the confusion and attempts to answer this question above.
Languages evolve slowly as products of their time, people who speak them, and the needs to describe both. Until very recently, English has not had need of a gender-neutral singular pronoun because societal attitudes did not require such. It was generally accepted that you were the gender/sex you biologically demonstrated at birth. Rare cases of ambiguous sex due to biological/developmental deformities aside, this is how most speakers of English were content to view the world. Binary usages like his/her, were not offensive to most speakers. And, in cases where people wanted an alternative, the most common uses were the singular they or one substitutions. Other past attempts at coining a new system were largely ignored, laughed at, or downright rejected as unnecessary.
The past few decades have seen quite a shift in our understanding of and attitudes towards human sexuality and gender identity. Homosexuality has been declassified as a mental disorder (it was listed as such as recently as DSMIII). Sexuality is no longer seen as a binary, tertiary or even quaternary state. We accept that people are heterosexual, homosexual, bisexual, pansexual, and myriad other orientations and preferences. What was once taboo to even mention in public can now be discussed airily on a sitcom...
Likewise, our societal understanding of gender has begun to evolve. The word transgender has entered the public consciousness in more than a small way. We're only beginning to adapt to the notion that people's gender exhibits similar fluidity to that of sexuality. And, as such, our language is still unequipped to properly handle this.
What will be the chosen gender-neutral singular pronoun? I'll be damned if I know. But, I do know we currently don't have one.
For now, unless the person in question directs me to do otherwise, I will continue to use the singular they until an alternative gathers enough steam to supplant it. Unfortunately, languages evolve far slower than society does, so I don't recommend that you hold your breath waiting.