If solo means walking up on your own with a map, then no, and it isn't a matter of advice. Park rules require every climber to be registered with a licensed operator and accompanied by a licensed guide. There's no permit for an unguided ascent.
Why the rule exists
Altitude sickness starts in your judgement. By the time you're making bad decisions you're usually the last to notice, which is exactly why a second pair of eyes isn't optional at 4,500 metres. The rule also means somebody knows where you are, and that a rescue can start from a known point rather than a search.
What solo actually looks like here
It means travelling alone and climbing with your own crew, which is a genuinely good way to do it. You get a guide to yourself, you walk at your pace, you stop when you want to, and nobody else's bad morning sets the schedule. Your crew will still be a handful of people, because one climber needs a guide, a cook and porters whether there's one of you or six.
The cost of being one
It's more per person, and the reason is arithmetic rather than policy. A guide's fee, a cook's fee and the park's crew charges are the same whether they're split one way or six, and there's usually a single-tent supplement on top. Expect a solo climb to cost meaningfully more per person than the same route for a pair, and ask for the breakdown so you can see where it goes.
Joining a group instead
Open-group departures put you with other climbers on fixed dates and bring the price down a long way. The trade is pace and company. You're walking with people you haven't met, and the group moves at the speed of whoever is having the hardest day. Some people make friends for life on those climbs. Some wish they'd paid for the private one.
Is it safe on your own?
About as safe as it is in a group, because the risks are altitude and cold rather than anything else. You're never alone on the mountain even when you book alone. If anything, a solo climber gets watched more closely, since there's nobody else for the guide to divide attention between.
Women climbing alone
Common, and normal. The crew are professionals and the camps are busy and public. If you'd rather have a woman guide, say so when you book rather than at the gate. There are fewer of them and they get booked early.
What a private climb changes day to day
More than you'd think. You start when you're ready rather than when six people are ready. You stop for photographs without feeling you're holding anybody up. If you have a bad afternoon, the itinerary bends around you instead of you trying to keep up with it. And you spend a week talking to your guide, which is how most people end up learning anything about the mountain.