Strategy Consulting Principal to PE in the GCC – Advice?

Hi everyone, I'm currently exploring a move from strategy consulting into private equity/ investing in the GCC and would appreciate perspectives from people familiar with the market.

Quick background: 11+ years of experience, currently at Principal level at a top strategy consulting firm in the Middle East, with previous experience at another MBB strategy consulting firm in Europe. Over the past several years, I've worked extensively with GCC sovereign wealth funds, investors and portfolio companies across investment strategy, commercial due diligence, portfolio strategy and value creation. I've also led work around investment platforms, target screening/ prioritization and investment cases.

I'm now looking to move to the principal investing side, ideally in the UAE, Qatar or KSA. I'm primarily considering PE funds, sovereign investment platforms and potentially other institutional investors where my background could be relevant.

Would be very interested in views from people who know the GCC PE market:

• How realistic is the consulting → investing transition at my level? 
• Which PE funds/ investment platforms in Dubai, Abu Dhabi or Riyadh are most receptive to senior strategy consulting profiles?
• Which headhunters are actually active in GCC PE/ SWF recruiting at the moment?
• Is networking directly with Partners/ MDs generally more effective than going through recruiters in this market?
• Any other advice on positioning my background or firms I should be targeting?

I've already approached a number of the larger global and regional PE recruiters, but the GCC market seems fairly quiet at the moment, so I'd be particularly interested in hearing from people currently in the region.

Happy to provide more context if useful. Thanks in advance.