Anything is possible.
Remove injector rail completely and have somebody start the engine. Then you can see the spray pattern. Put something under it to catch the fuel. However, in my opinion this is not likely as the engine starts and runs o.k. when warm.
Have you measured fuel pressure? How old is the fuel filter?
Have you ran the fuel tank completely empty lately?
Can you measure voltage of coolant and intake air temperature with ignition on, engine off when cold? (Backprobe at sensor or ecu connector or use breakout box).
And the same for the map-sensor signal?