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TPS7A7002: unpredictable shutdown

Part Number: TPS7A7002
Other Parts Discussed in Thread: LM2596, TPS54560

Greetings, I have a problem with TPS7A7002, I hope somebody can explain me, why it behave like that. I have a device with GSM modem (SIMCom SIM800 series), which requires from 3.4V to 4.4V power supply, which can withstand current bursts up to 2A (duration about 1ms). For a long time it worked well, being powered by dirty cheap chinese copy of LM2596 (it's a 3A dc/dc regulator), from 24V input. And now I decided to rebuild the PCB, to make it much smaller, and instead of dc/dc, I used TPS7A7002, powered from 5V. And here is the tricky part of the story - my GSM modem became restarting at the moment of connection to network. There are oscilloscope plots during connection with old dc/dc:

and new TPS7A7002:

As we can see on the screenshot, the're is huge drop, down to 2.8V, and this is the reason of GSM modem's reset. So, the main question is - do TPS7A7002 has some kind of short circuit protection, not mentioned in datasheet, or its current-sensing circuit is SO sensitive, that 3.36A limit isn't enough? And is there any possible to use LDO in this case, or I should go back to dc/dc?

P.S. here is schematic and part of PCB

(adding 470 uF capacitors in parallel to C4 and C5, won't change nothing)

Could somebody help me, please?