Car starts intermittently

[LoveMyLexus1]

Car sat for 10 weeks because I couldn't drive it.  Now it intermittently starts.  I can crank it  a few times, then 2 hours later or so it will start, but not always.  When it does start during this scenario, at first it acts like it won't start then suddenly roars to life as if it got a sudden gush of gas.  

I've also had it fail on me while at the grocery store.  The temperature had dropped 20 degrees or so, but I had just filled it up with Shell gas.  It seems to be less likely to start when the fuel tank gets about halfway full.  It also seemed to be less likely to start during a cold snap.

I left it and called an Uber.  Scheduled a tow for the morning, but when I got there, it started.

Another time I did have it towed, but it started perfectly at the shop.  I left it there 3 days hoping it would fail.  They say diagnostics can't be done until it fails.

It seems to me that maybe fuel isn't getting to the injectors.  If it's warm, the fuel expands and it's then available to be pulled into the EFI system.

Note:  I bought this car in 2013.  It had about 60,000 miles on it.  The battery is 1 year old.  I change the synthetic oil every 3 months or 5000 miles.  The starter turns over just fine, so it's not the alternator.  This is the first real problem I've had with it.  Replaced the timing belt and water pump at 88,000 miles.  It now has about 98,000 miles on it.  

I could sure use some ideas about how to diagnose the problem.  If anyone can help, I would appreciate it.

Edited October 11, 2021 by LoveMyLexus1
Added important information.